According to the Sun, both Spurs and Aston Villa are interested in signing the Australian international, who has been in fine form as Fulham push for a place in next season's Europa League.
Schwarzer has one year left on his current contract and would bring with him a wealth of experience should he leave Craven Cottage and sign for either Harry Redknapp or Martin O'Neill.
Meanwhile, Djibril's Cisse's proposed move to White Hart Lane has taken a step closer with the player's agent, Ranko Stojic, claiming a deal is already in place.
Cisse is reportedly unsettled at Sunderland and was dropped to the bench for last weekend's draw at Bolton.
Sunderland manager Ricky Sbragia is set to further alienate the Frenchman by naming Kenwyne Jones as a lone striker for Monday's Premier League clash at Portsmouth, and Cisse's agent said: "We have an agreement with Tottenham to sign a 12-month contract."
As well as Schwarzer and Cisse, Redknapp could also be joined at White Hart Lane by his son, Jamie, who has been offered a coaching role next season.
Redknapp Jnr used to play for Tottenham before a knee injury forced him to retire in 2005.