The Manchester Evening News reports that the Cottagers are keen to take the 25-year-old on loan from United when the January transfer window opens.
New Fulham head coach Rene Meulensteen is reportedly an admirer of the player and is keen to be reunited with the Brazilian at Craven Cottage.
Anderson has made just two starts for United this season.
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Liverpool have unveiled their new home kit for next season, sticking with the hugely popular retro styling of the current shirt.
The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.