"(The) Justice Department and my national security team will continue to cooperate with Turkish authorities to determine how we will make sure that those who carried out these activities are brought to justice," Obama said.
Since an attempt to overthrow him failed in July, Erdogan has taken harsh tactics to consolidate power, worrying human rights advocates and US officials alike.
The Turkish leader was expected to press Obama on US-based cleric Fethulah Gulen
, under suspicion in Turkey for helping plot the coup. Gulen denies involvement, and the US is demanding evidence of his participation, but rejecting the request could worsen ties between the two countries.
Erdogan said his government was preparing to submit additional evidence about the coup to the United States, and would soon send his justice and Interior ministers for consultations with American authorities.
For the US, alienating Erdogan remains an unwelcome prospect given Turkey's critical role in the battle against ISIS in Iraq and Syria. Even on that front, disagreements persist. Washington has pressed Ankara to end airstrikes on Syrian Kurds, whom the US is backing in the fight against ISIS terrorists.
It's the second day of high-stakes diplomacy for Obama, who arrived here Saturday to an inauspicious welcome: no red-carpeted stairs for Air Force One and open quarreling on the tarmac between Chinese and US officials over press access. Other leaders arriving for the G20 here were greeted with a far grander welcome.
The rancor continued throughout Saturday as Obama met with Chinese President Xi Jinping to cement a carbon reduction agreement
and haggle over longstanding disputes like cybersecurity and maritime aggression.
Obama said Sunday the US would never apologize for its insistence upon press access, but said he understood why Chinese officials may have chafed at the Americans' demands.
"We don't leave our values and ideals behind when we take these trips. It can cause some friction," Obama said following his meeting with Theresa May, the newly installed British prime minister. At the same time, Obama cautioned against inflating the incident.
"I wouldn't over crank the significance of it," he said. "We've got a lot of planes and helicopters and a lot of cars and a lot of guys and, you know, if you're a host country, sometimes it may feel a little bit much."
Haggling over Syria
Obama was beginning his day of talks as his Secretary of State John Kerry was working to reach a deal on ending violence in Syria. The President said Kerry and Russian foreign minister Sergey Lavrov had been working "around the clock" to hammer out an agreement, but hadn't yet reached a final accord.
"I think it's premature for us to say that there is a clear path forward, but there's the possibility at least for us to make some progress on that front," Obama said.
Obama and Russian President Vladimir Putin were expected to interact during this week's Group of 20 summit, though a formal meeting was not on the books. Any deal to end widespread suffering in Syria was likely to be met with doubt after past settlements failed -- a fact Obama acknowledged on Sunday.
"Given the failure of previous cessations of hostilities to hold, we approach it with some skepticism," Obama said.
But he said any work toward easing the deeply troubling humanitarian crisis was valuable.
"It is worth trying," he said. "To the extent that there are children and women and innocent civilians who can get food and medical supplies and, you know, get some relief from the constant terror of bombings, that's worth the effort."
Deep divides over Syria have marred Obama's relationship with Putin, adding to a litany of discord between the US and Russia that's driven relations to their lowest level since the Cold War.
This week's interaction between Obama and Putin could be a final attempt to salvage what's become one of the most acrimonious relationships on the global stage. Putin's persistent support for the Syrian regime, Moscow's moves in Ukraine and the charge that Russia may be meddling in the US presidential contest have built a deeply antagonistic dynamic between the two leaders.
A deal on Syria could place US-Russia ties in a firmer position as Obama prepares to leave office. American diplomats have been pressuring Moscow to use its influence in Damascus and force the Syrian regime to end its strikes on civilians and US-backed opposition fighters.
Obama said on Sunday that Russian participation was essential.
"Our conversations with the Russians are key because, if it were not for the Russians, then Assad and the regime would not be able to sustain its offensive," he said.
Obama used a morning session with British Prime Minister Theresa May to gauge the new leader's plans for executing Britain's exit from the European Union, a move the US has insisted must proceed in an organized fashion.
Obama said Sunday the US-UK "special relationship" would persist as Britain works to exit the European Union.
"The bottom line is that we don't have a stronger partner anywhere in the world than the United Kingdom," Obama said. "Despite the turbulence of political events over the last several months, we have every intention to making sure that that continues."
It's Obama's first face-to-face encounter with May since she replaced David Cameron, a close Obama ally who resigned in the aftermath of the Brexit vote. Obama took the unusual step of personally lobbying against Britain's exit from the EU, but if the aftermath of the vote has said he would stand by Britain amid the country's internal struggles.
May insisted during a joint appearance with Obama that exit negotiations would move forward.
"Brexit does indeed mean Brexit," she said.