Dembele's contract at Craven Cottage ends in the summer, although he will be sent back to the Championship club on loan until the end of the season.
The 19-year-old Frenchman joined Fulham from Paris St-Germain in 2012 and has scored 11 goals this campaign.
Spurs already have unrelated 28-year-old Belgium midfielder Mousa Dembele in their squad.
Fulham, meanwhile, have signed defender Michael Madl, 27, on loan until the end of the season from Austrian side Sturm Graz.
The Whites, who are 19th in the Championship, only four points above the relegation zone, and have only won one of their last 15 matches, have an option to make the deal permanent.
Dembele, who has represented France at under-20 level, was only 17 when he made his Premier League debut in November 2013, although Fulham were relegated from the top flight in that season.
Tottenham are fourth in the Premier League, five points clear of fifth-placed Manchester United, and aiming to qualify for the Champions League for only the second time.
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Fulham striker Moussa Dembele is having a medical and is set to join Tottenham in a transfer worth a reported £5m.