Luis Videgaray also told "Anderson Cooper 360˚" that his nation was very willing to discuss the North American Free Trade Agreement (NAFTA) with the White House.
Videgaray said the conversation last Friday between Trump and Peña Nieto was "very constructive." The two leaders agree on more points than they disagree on, he said Thursday.
"It was not a confrontational call," he said.
The presidents agreed to keep working on important matters such as security. As to conflicting reports about issues raised in the call, the foreign minister said, based on his recollection, CNN's reporting of Trump's comments about offering US troops to help with drug cartels
According to an excerpt of the transcript of the call with Peña Nieto provided to CNN, Trump said, "You have some pretty tough hombres in Mexico that you may need help with. We are willing to help with that big-league, but they have be knocked out and you have not done a good job knocking them out."
The excerpt of the transcript obtained by CNN differs from an official internal readout of the call that wrongly suggested Trump was contemplating sending troops to the border in a hostile way. An Associated Press report said Trump also threatened to send US troops to stop criminals in Mexico unless the government did more to control them.
"That was ... nothing close to reality," Videgaray said.
He said media reports from a Mexican journalist that Trump told Peña Nieto that Mexico was going to pay for a border wall whether they like it or not was false.
"It's very clear, we have a significant difference and the presidents have a significant difference," the foreign minister said. "And it's ... well-known, but they agreed that they would not continue to publicly argue about that, because there are so many other things that we need to work on."
Videgaray said he talked to US Secretary of State Rex Tillerson on Thursday and the two planned to meet in their new roles soon.
NAFTA 'crafted in a different world'
As to NAFTA
, Videgaray said, "We're absolutely open to making NAFTA better."
He pointed out the agreement -- which also includes Canada -- was signed in 1994.
"It was crafted in a different world," he said, adding that there was no ecommerce at the time. Before meetings can be held between the nations, Mexican leaders must consult with their country's Senate, union bosses and businesses, he told CNN.
"This does not have to be a negotiation in which one of the countries loses," he said. "We can make it a very good deal for both countries."
Peña Nieto had canceled a meeting with Trump that had been set for this week after renewed tensions erupted last week over Trump's plan to build a wall on the border.
"Unless Mexico is going to treat the United States fairly, with respect, such a meeting would be fruitless and I want to go a different route," he said in prepared remarks. "We have no choice."
Trump had tweeted that it would be better to skip the meeting if his Mexican counterpart continued to insist Mexico would not pay for the wall.
While saying that Mexico paying for the wall is "totally unacceptable," Videgaray said his ministry would take a cooperative stance toward Mexico's northern neighbor and largest trading partner.