Nigeria House of Representatives yesterday voted to offer Mr Jammeh asylum to help negotiations.
A motion passed in the House of Reps gave members' backing to regional efforts to resolve the dispute, sparked by Jammeh's refusal to accept election results.
The Reps said "the clock is ticking fast" for The Gambia and there was a need to step up diplomacy, as the possibility of violence and mass displacement threatened West African stability.
They called on Jammeh to "respect the will of the people", who voted for opposition candidate Adama Barrow in the December 1 election.
President Muhammadu Buhari, who is leading the regional diplomatic effort, should "extend Nigeria's readiness to offer... Jammeh safe haven in Nigeria to live securely as a way of ending the political stalemate in The Gambia", they added.
Mr Jammeh's term officially ends next Wednesday, but he wants the results annulled after the electoral commission changed some - though it insists this did not affect the final outcome.
The 51-year-old leader, who initially accepted defeated in the 1 December poll, seized power in the tiny country in 1994 and has been accused of human rights abuses, although he has held regular elections.
The Supreme Court is unable to hear the challenge until May because of a shortage of judges, and Mr Jammeh says he will not step down until then.
Foreign leaders from regional bloc Ecowas, led by Nigerian President Muhammadu Buhari, are due to arrive later on Friday in The Gambia to again try to persuade Mr Jammeh to step down on schedule.
The grouping has warned that it would consider removing him using military force if he refuses.
But Mr Barrow, a property developer, told the BBC's Newsday programme that he would prefer a "peaceful transition".
He said he welcomed the move by Nigeria Reps to offer Mr Jammeh asylum, but said he did not feel the situation would "get to this level".
"We want to keep Jammeh in The Gambia, I don't think there's any need for him to go to another country."
He called on Mr Jammeh to "respect the constitution" and engage in direct talks.
"We solve our problems within ourselves without the intervention of anybody. I think that's what we'd prefer," he said.
A lawyer for Mr Jammeh on Thursday filed a request with the Supreme Court asking for an injunction to block Mr Barrow's swearing-in. Mr Barrow won 43.3% of the vote compared with Mr Jammeh's 39.6%. A third candidate, Mama Kandeh, got 17.1%.
Nigerian Foreign Minister, Geoffrey Onyeama, said on Tuesday that "violence should be avoided but nothing is ruled out" by regional bloc ECOWAS to ensure The Gambia's constitution was upheld. Jammeh has taken legal action against the election result and said he will not step down until his complaint is heard.
That has raised the prospect of months of political deadlock because The Gambia lacks Supreme Court judges to handle the case.
Nigeria has previously given asylum to a number of African political leaders, including the Liberian warlord-turned-president Charles Taylor.
Dr. Yousef Al-Othaimeen who has been concerned about the electoral dispute and political impasse following the December 2016 presidential election in the Gambia, urged the President to avoid any act that may endanger the peace and stability of the country, which will chair the next Islamic Summit. He also appealed to President Jammeh to ensure a peaceful transfer of power to the President-elect Adama Barrow on Jan 19, 2017 in accordance with election regulations of the country.
The OIC delegation was composed of Amb Samir Bakr and Amb. Yahya Lawal who delivered the message on Jan 9 2016 in Banjul in the presence of Ministers of Foreign Affairs, Tourism and Health.