President Joe Biden waits for the start of a lunch with the Group of Seven leaders at the Schloss Elmau hotel in Elmau, Germany, Monday, June 27, 2022, during the annual G7 summit. AP
Analysts believe the meeting could play a crucial role in breaking down Turkey's resistance to bids by Sweden and Finland to join the Western defence alliance in response to the war.
The two leaders have had a chilly relationship since Biden's election because of US concerns about human rights under Erdogan.
Biden and Erdogan last met briefly in October on the sidelines of a G20 summit in Rome.
"We spoke with Mr Biden this morning and he expressed his desire to get together tonight or tomorrow. We said it was possible," Erdogan said.
He was speaking to reporters before flying to Madrid for talks that will start with his meeting with the leaders of the two Nordic countries and the NATO secretary general.
Erdogan said he wanted to see the results of preparatory talks held on Monday in Brussels before deciding whether Sweden and Finland had done enough to lift his objections to their membership of the military alliance.
Turkey is a NATO member and could veto both countries' applications at the summit.
"We are a 70-year-old member of NATO. Turkey is not a country that randomly joined NATO," Erdogan said.
"We will see what point they (Finland and Sweden) have reached," he added. "We do not want empty words. We want results."
- 'Open mind' -
Ankara has accused Finland and more particularly Sweden of offering a safe haven to Kurdish militants who have been been waging decades-long insurgency against the Turkish state.
The Turkish leader has also called on the two countries to lift arms embargoes imposed on Turkey in 2019 over Ankara's military offensive in Syria.
The two countries went into the NATO meeting open to the possibility that Turkey might only lift its objections after the summit concludes on Thursday.
"We have made progress. That is definitely the case," said Swedish Foreign Minister Ann Linde.
"We are prepared for something positive to happen today, but also for it to take more time," she added. "We must be patient and continue discussions even after the summit."
Finnish President Sauli Niinisto said that he was neither "optimistic nor pessimistic at this stage".
"We will go into this afternoon's discussion with Erdogan with an open mind," he told Finnish media.
- Fighter jet talks -
Erdogan's ability to maintain a close working relationship with Russian President Vladimir Putin while supporting Ukraine's war effort has made him an important player in the conflict.
But those ties have also complicated his relations with Biden and the NATO bloc.
Washington has sanctioned Ankara for taking delivery of an advanced Russian missile defence system in 2019.
The purchase saw the United States drop Turkey from the F-35 joint strike fighter programme and impose trade restrictions on its military procurement agency.
But Washington has signalled that it may be willing to move past the dispute.
Biden's administration has dangled the possibility of supplying Ankara with older-generation F-16 jets that could replenish Turkey's ageing air force fleet.
"The most important issue is the F-16 issue. It is still on the table," Erdogan said of his upcoming talks with Biden.