Arsene Wenger had previously made a £2m bid for 37-year-old Schwarzer, which was turned down twice - despite the keeper lodging a transfer request.
The Sun reports that the Gunners have now upped their offer to £4m in an attempt to secure Schwarzer's services for their latest title attempt.
More: Arsenal transfer news and rumours
Fulham boss Mark Hughes will be reluctant to let Schwarzer go, unless he is able to complete a rumoured deal with Manchester City for the out-of-favour Shay Given.
Hughes has also been linked to £2m-rated Reading goalkeeper Adam Federici, 24.
Arsenal, who currently rely on Manuel Almunia between the sticks, have reportedly considered moves for Lille's Mickael Landreau, and Lazio's Fernando Muslera is also a target.
