The Cottagers want Sterling - and the veteran wants to see the move happen.
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However for now, Chelsea are yet to set their terms for Sterling's departure, despite wanting to move him on.
The Mirror says Fulham are happy to wait until Monday's deadline day with the aim of taking Sterling off Chelsea's hands for as low a fee as possible.
A loan arrangement is the most likely option for Fulham, which are in a good position knowing Sterling favours remaining in L:ondon this season - and with Chelsea eager to clear him off the books.