The Aussie keeper has been targeted by Arsene Wenger as a short-term solution to Arsenal's problems in goal.
Schwarzer, 37, has a year left on his contract with Fulham and is aware that Arsenal want him to replace Manuel Almunia.
The Fulham keeper was exceptional during the club's run to the Europa League final against Atletico Madrid.
Hodgson is reluctant to allow the keeper to leave Fulham, but does not want to stand in the way of a dream move.