Arsenal Manager Arsene Wenger looks set to win his battle for - after going back with a fresh £3.5million bid.
Aussie keeper Schwarzer, 37, is aware of Arsenal's interest and has told Fulham he wants to quit Craven Cottage.
It has left Fulham boss Roy Hodgson resigned to losing Schwarzer as Arsenal close in on their top keeper target.
Wenger sees Schwarzer as the experienced keeper he needs to steady his defence and also not block the progress of his youngsters coming through the ranks.
Spanish first choice Manuel Almunia, who was axed at the end of the season with the club claiming he had a wrist injury, will be allowed to go but they value Polish youngster Wojciech Szczesny, 20, as a huge prospect for the future.
Wenger sent scouts to look at Schwarzer at the end of the season and has watched him personally in South Africa.
Fulham have knocked back one bid from Arsenal for around £3m but the Gunners keep going back and now the Cottagers are resigned to losing him.