Regional armies, who entered Gambia late on Thursday, were meanwhile poised to remove him by force if required, as even his army chief, who had stood beside the former coup leader, recognised his rival Barrow as commander-in-chief.
West African leaders Alpha Conde of Guinea and Mauritania's President Mohamed Ould Abdel Aziz travelled to the capital Banjul on Friday to allow Jammeh one last chance to cede power peacefully.
"I can assure you that he has agreed to leave," Mai Ahmad Fatty, Barrow's special advisor, told Reuters in Senegal's capital Dakar. He could not say where Jammeh would go into exile.
Barrow, who won the Dec 1 poll by a slim margin, was sworn into office at Gambia's embassy in Dakar on Thursday and immediately called for regional and international support.
A midday deadline was extended on Friday as negotiations, which diplomats said were focusing on a deal that would grant Jammeh immunity from prosecution, continued.
Final act
One regional diplomat said the delegation was planning to spend the night in Banjul and Jammeh was looking for "extremely solid guarantees" before leaving.
"There is a real possibility this could work. I don't think he is going the (Saddam) Hussein route," he said, referring to the Iraqi leader who was arrested in 2003 following an invasion, then tried and hanged.
Jammeh, in power since a 1994 coup, initially conceded defeat to Barrow before back-tracking, saying the vote was flawed and demanding a new ballot.
During his 22-years in power, Jammeh, who once vowed to rule Gambia "for a billion years", has been accused by rights groups of torturing and killing perceived opponents. And some Gambians were angered by the prospect of granting him immunity.
"I don't want that man to escape punishment from us," said Lamin Darboe, 35, a Gambian shopkeeper who was present at Barrow's speech in Dakar on Friday. "Wherever he moves to we'll follow him there and grab him."
Thousands of tourists, who'd flocked to the popular beach holiday destination for a break from the harsh European winter, also left this week.