Schwarzer, 37, is expected to complete a switch across London to The Emirates following the end of Australia's World Cup adventure.
The former Middlesbrough custodian will cost the Gunners around £3.5million, according to the News of the World, and the Cottagers will put that fee towards a deal for England blunder keeper Green.
The 30-year-old Hammers stopper was dropped by coach Fabio Capello following his shocking mistake in the 1-1 World Cup draw with the USA - but Fulham see him as an ideal replacement for the departing Schwarzer.
New Irons boss Avram Grant is open to the idea of selling Green, as he is believed to be keen on a swoop for Stoke stopper Begovic.
Grant knows Begovic from his Portsmouth days - but Tony Pulis will not want to sell a player he sees as the future number one at the Britannia Stadium.