Jammeh has made clear he will not step down until the country's Supreme Court decides on his legal challenge seeking to annul the results.
Nigerian President Muhammadu Buhari is leading the three-nation delegation that includes Ghana's former president John Mahama and Liberian leader Ellen Johnson Sirleaf, who attempted similar negotiations last month without success.
Jammeh has made clear he will not step down until the country's Supreme Court decides on his legal challenge seeking to annul the results of last month's polls, which he initially conceded losing.
On his his way out of a luxury hotel to meet Jammeh, Buhari told journalists "only God knows" if the Gambian leader will step aside. But Nigerian foreign minister Geoffrey Onyeama said he was "pretty optimistic that the talks will not fail this time."
The delegation landed with just five days left of Jammeh's five-year term to go, and with a warning by the president late Tuesday to the international community that "undue external interference" was unnecessary.
The Supreme Court is unlikely to sit and hear his legal challenge before May, ratcheting up tensions with the winner of the December 1 election, Adama Barrow, whose inauguration is due January 19.
The prolonged political uncertainty and fear of unrest has pushed thousands of Gambians across the border into neighbouring Senegal and further afield to Guinea-Bissau.
Tibna Sambe Na Wana, the national coordinator for Guinea-Bissau's refugee commission, said more than 1,000 Gambians had crossed into the country, where they do not require a visa, in recent days.
"It is clear that the total number is far higher than a thousand and rising daily," Na Wana said.
Women, children and the elderly made up the greatest numbers, the official said, with more than 500 passing one border post near the town of Jegue in three days.
"They say they are scared of a military escalation," Na Wana added.
In Senegal, the UN's refugee agency said "several thousand people" had crossed into the southern Casamance region from The Gambia, especially children.
Senegal and The Gambia have deep ethnic and linguistic ties, and most families have relatives living across the border.
"Most arrivals in Senegal are Gambians and Senegalese who have been working or living in The Gambia," said Liz Ahua, UNHCR's representative for West Africa in Dakar, but said Africans from several other nations were also crossing.
With the migrants moving into their relatives' homes, households have doubled or tripled in size in a few days, the UN agency said, putting a strain on food supply.
A nation of fewer than two million people, The Gambia already accounts for the highest number of migrants per capita of any nationality crossing the Mediterranean on smugglers' boats to Italy.
The effects of the crisis were also being scrutinised across the continent in Addis Ababa, where the African Union (AU) declared it would no longer recognise President Jammeh as head of state on January 19, regardless of the Supreme Court case.
Citing "the inviolable nature of the outcome of the presidential elections held on 1 December 2016 in The Gambia," the AU called on Jammeh to respect the constitution and cede power to Barrow on that date.
Jammeh has said he wants to wait for the Supreme Court to sit and hear his case, but the Gambian Bar Association said Friday his term could not legally be extended except through a referendum altering the constitution.
The president has few allies and has faced almost universal condemnation for clinging to his post, though Nigerian MPs said Thursday the country should consider offering him asylum.