(CNN) -- Russian President Vladimir Putin phoned his U.S. counterpart, Barack Obama, on Friday to discuss the tenuous situation in Ukraine -- the latest exchange between two leaders who have been at loggerheads about what's happened and what should happen next.
According to the White House, Putin called to talk about an American proposal "for a diplomatic resolution to the crisis" and the two presidents agreed that their respective top diplomats "would meet to discuss next steps."
The back-and-forth also gave Obama the opportunity to express, as he's done repeatedly in recent week, his opposition to what he described as Russia's taking over of Crimea, which just a few weeks ago was part of Ukraine. He also raised concerns about a reported massing of Russian troops near the border, which has raised fear of further incursions.
"President Obama made clear that this remains possible only if Russia pulls back its troops and does not take any steps to further violate Ukraine's territorial integrity and sovereignty," the White House said.
The phone call came on the same day an interview aired in which Obama said Russia could ease tensions with Ukraine if it moved its troops back and begin direct talks with the Kiev government
"You've seen a range of troops massing along that border under the guise of military exercises, but these are not what Russia would normally be doing," the President said in the CBS News interview. "It may simply be an effort to intimidate Ukraine, or it may be that they've got additional plans.
"And, in either case, what we need right now to resolve and de-escalate the situation would be for Russia to move back those troops and to begin negotiations directly with the Ukrainian government as well as the international community."
Russia may have 40,000 troops near its border with Ukraine, two U.S. officials told CNN on Thursday. The officials said that this estimate was largely based on satellite imagery and that a firm number is difficult to assess.
However, a spokesman for Ukraine's Council of National Security and Defense, Yarema Dukh, told CNN his government estimates 88,000 Russian troops are at the Ukrainian border.
U.S. officials said they believe the higher estimates may reflect Russian troops on alert farther to the east.
Russia has said its troops are carrying out snap military exercises in the region.
Also Friday, U.N. Secretary-General Ban Ki-moon reported on his talks with Putin and Ukraine leaders, saying he asked Putin if any military action was about to happen.
"This is what the President Putin told me: That he had no intention to make any military move," Ki-moon said. "I should also tell you that, at the same time, President Putin also expressed his concern about some extreme radical elements and as such along the borderline."
Ki-moon said diplomacy is the only way.
"At this time of heightened attention, even small sparks can ignite larger flames of unintended consequences," he said.
Obama said he believes Putin still harbors a grievance over the 1991 dissolution of the Soviet Union. "You would have thought that, after a couple of decades, there'd be an awareness on the part of any Russian leader that the path forward is not to revert back to the kinds of practices that were so prevalent during the Cold War."
Instead, he said, Putin should be moving toward further integration with the world economy.
Obama said that Putin has described the breakup of the Soviet Union as tragic, and that he may feel that the West has taken advantage of Russia.
"He may be entirely misreading the West," Obama said. "He's certainly misreading American foreign policy. We have no interest in encircling Russia, and we have no interest in Ukraine -- beyond letting the Ukrainian people make their own decisions about their own lives."
He rejected the notion that Russia's concerns over maintaining its influence along its borders might justify its invasion of other countries.
"Certainly, they're going to have influence -- because of trade and tradition and language and heritage -- with Ukraine," he said. "But there's a difference between that and sending in troops and, because you're bigger and stronger, taking a piece of the country. That is not how international law and international norms are observed in the 21st century."
Russia rejects U.N. resolution
Obama's comments were broadcast as Russia dismissed a U.N. General Assembly resolution that branded Crimea's secession referendum invalid, calling it "counterproductive."
The Russian Foreign Ministry said the U.N. vote -- which followed Moscow's annexation of Ukraine's Crimea region last week on the basis of the hastily called referendum -- would do nothing to help resolve the situation.
"The counterproductive initiative with the General Assembly's resolution only complicates efforts to stabilize the internal political crisis in Ukraine," the ministry said in a statement.
The U.N. vote, held Thursday, saw 100 countries back the nonbinding resolution, with 11 opposed and 58 abstaining.
Russia's actions in Ukraine have been widely condemned by the West and prompted concern in Kiev and other former Soviet states that further incursions may follow.
On Friday, Russian Foreign Minister Sergey Lavrov told reporters in Moscow, "I hope that the situation will be accepted with a sense of reality by our European neighbors."
Also in Moscow, Russian Defense Minister Sergei Shoigu said that Ukrainian troops in Crimea who had chosen to remain in the Ukrainian Armed Forces had withdrawn. "The change in state symbols on all ships and in all divisions that have sided with the Russian army has been completed," he said.
A spokesman for the Ukrainian Defense Ministry in Crimea, Vladislav Seleznev, said military personnel and their families from a number of military bases were expected to depart Saturday for inland Ukraine.
Putin said that events had demonstrated that Russian forces were prepared. "The recent events in Crimea were a serious test," the Russian President said at a ceremony marking military promotions, according to the Russian news agency Interfax. "They demonstrated both the completely new capabilities of our Armed Forces and the high morale of the personnel."
On Friday, Ukraine's ousted President Viktor Yanukovych stepped into the fray with a call for Ukrainians to demand a referendum on the future status of each region in the country, according to Russian state news agency ITAR-Tass.
"As the President whose thoughts and heart are together with you, I call on each reasonable citizen of Ukraine -- don't let the imposters use you! Demand a referendum on the determination of the status of each region within Ukraine," Yanukovych said in an address to the Ukrainian people.
He said that a referendum for all the regions was the only way to stabilize the country, and that early presidential elections planned for May will be neither fair nor constitutional.
Yanukovych also said he wished to be removed from his post as chairman of the Party of the Regions.
Yanukovych resurfaced in Russia days after leaving Ukraine amid bloody street protests in February against his decision to turn away from a European Union trade deal in favor of closer ties with Moscow.
He and Russia say he is still the legitimate leader of Ukraine, but the West disagrees. The interim government in Kiev was voted in by a large majority in parliament, including members of Yanukovych's own party.
Russia seized control of Crimea amid the political upheaval that followed Yanukovych's ouster -- a step cemented by the controversial referendum and a new treaty that absorbed the Black Sea peninsula into the Russian Federation.
Now, observers in the West fear Moscow may make incursions into eastern Ukraine, where there are strong ties to Russia, despite the threat of deeper EU and U.S. sanctions.
In another development Thursday, former Ukrainian Prime Minister Yulia Tymoshenko, who was released from prison last month, said she intends to run for president in elections on May 25.
After more than two years in prison, she was released in February after the ouster of her archrival, Yanukovych.
Tymoshenko said she intended to ask delegates at her Batkivshchyna Party congress on Saturday to nominate her as a presidential candidate.
Ukraine's elections are taking place against a backdrop of poor economic conditions, Moscow's annexation of Crimea and rumblings of discontent in the mainly Russian-speaking eastern regions.
Tymoshenko's announcement came as the International Monetary Fund announced a $14 billion to $18 billion bailout for Ukraine to avoid bankruptcy. The bailout is tied to painful reforms as the country faces an escalating standoff with Russia.
Ukraine Foreign Minister Oleksandr Shlapak said he was expecting the first transfer to be in the amount of $3 billion, though he did not say when.
Obama on Thursday praised the loan agreement between the IMF and Ukraine, and said the United States will do its part to help.
The U.S. Senate on Thursday approved $1 billion in loan guarantees for Ukraine, along with sanctions against Russia for its intervention in Ukraine.
Earlier this month, the House of Representatives approved Ukraine loan guarantees and is now voting on sanctions.
A large majority of Crimeans voted in favor of leaving Ukraine for Russia in the referendum held 12 days ago. It was dismissed by the interim government in Kiev and the West as illegitimate.
The Black Sea peninsula, which has a majority ethnic Russian population, was part of Russia until 1954 and has long historical and cultural ties to the country.
CNN's Greg Botelho, Marie-Louise Gumuchian, Radina Gigova, Brian Walker, Alexander Felton and Victoria Butenko contributed to this report.