The right-back, who has 33 caps, has reportedly moved from the Championship club for around 15 million pounds ($18.90 million) and will wear the number 21 shirt.
Castagne started his career at Belgian Pro League side Genk before moving to Serie A side Atalanta in 2017. He joined Leicester in 2020 and helped them win the FA Cup in 2021.
"It's been a long transfer window, but finally I'm here, and I'm happy to be here," Castagne, who scored five goals with eight assists in 112 games at Leicester, said in a statement.
"It's a big relief. I've talked to the manager (Marco Silva) and I just can't wait to start training and playing."
Fulham host Tottenham Hotspur in the League Cup second round on Tuesday before a league trip to Manchester City on Saturday.