Adama Barrow made the announcement on Twitter, after hours of talks aimed at persuading Mr Jammeh to recognise the result of the presidential election.
West African nations have deployed troops in The Gambia threatening to drive him out of office by force.
Mr Barrow has been in neighbouring Senegal for days. He was sworn in in the Gambian embassy there on Thursday.
His legitimacy as president has been recognised internationally, after he won last month’s elections.