He will take over on 30 December, with caretaker boss Ray Lewington remaining in charge for Saturday's away game against Birmingham City.
Fulham sacked Lawrie Sanchez last week with the club in the Premier League's bottom three.
"I am determined that we will overcome the current position," Hodgson, 60, told the club's official website. "I understand that it will be a challenge, but one that I will relish."
Fulham's new manager will be at their game at St. Andrews on Saturday before taking training with the Craven Cottage squad on Sunday.
His first game in charge will be the local derby at home against Chelsea on New Year's Day as he tries to arrest a dramatic slump in form for the Cottagers.
"After long talks with the chairman and the board I am convinced that we can take the Club to a new level," said Hodgson.
But it proved to be no more than speculation and Hodgson will be unveiled by Fulham at a news conference on Monday.
"It is my vision to combine the best of the fantastic traditions of the English Premier League with the flair and innate professionalism of the continental game, which I hope will return Fulham to its tradition of playing attractive passing football," said club chairman Mohamed Al Fayed.
"I believe that Roy has immense experience in all areas of football and can drive this club forward for the long term.
"My commitment to this club has never dwindled and I am more than ever determined that we have the right man to turn round our current predicament and build for a successful future."