Sessegnon, who is in the final 18 months of his deal, has been in talks since last year.
Both Tottenham and Manchester United have shown strong interest in Sessegnon, 18, who can expect a substantial pay rise.
Fulham vice chairman Tony Khan and chief executive Alistair Mackintosh are handling the contract discussions and Ranieri hopes to hear positive news soon.
“Sooner or later his agent will find a solution,” said the Fulham manager. “It is not my job. I want him, I believe in him. I hope soon it can be finished.”
Sessegnon, who signed a professional deal at Craven Cottage when he was 17, is one of the brightest prospects in English football and has been a rare shining light for Fulham this season.
Ranieri has praised Sessegnon’s impeccable attitude but he has urged the teenager to develop a nasty side to his game as Fulham gear up for a fight against relegation.
“I want him to be more aggressive,” said Ranieri ahead of tomorrow’s clash at Burnley.
“Sometimes in the duel he is shy. I do not want to say a nasty word in Italian but I want him to be more nasty.”
Ranieri, meanwhile, has warned Fulham it is crucial they sign an “experienced leader” during the January window.
Fulham have conceded 47 goals in 21 matches and Ranieri said: “We have to bring somebody to survive. It would be a boost for the players.”