Schwarzer, 37, is expected to complete a switch to the Gunners following the end of Australia's World Cup journey.
The former Middlesbrough and Bradford stopper will cost Arsenal around £3.5m, according to the News of the World, and the Cottagers will put that fee towards a deal for England's Green.
The 30-year-old was dropped by coach Fabio Capello following his shocking howler in the 1-1 World Cup group 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's Begovic.
Source: Team Talk