Sahin joined Real from Borussia Dortmund last summer, but the Turkey international struggled to make an impact during his debut season with the Spanish champions.
His prospects for this term again look bleak and the 23-year-old has been strongly linked with a move to the Premier League.
He has already dismissed talk he could join Tottenham Hotspur as part of Real’s bid for Luka Modric, while he is rumoured to be a target for Liverpool.
Arsenal, though, are reported to be favourites to sign Sahin after Arsene Wenger confirmed the midfielder is someone he is following.
With Alex Song set to join Barcelona, Wenger could step up his interest in Sahin after admitting he may make another signing before the transfer window closes.
Mourinho remains prepared to let Sahin head out of the Santiago Bernabeu on a temporary basis, but does not want to ditch him permanently.
“The most important thing is that he is happy,” said Mourinho. “The best for him and for Madrid is that he goes out on loan.
“Madrid don’t want to sell him. He needs to play and to prove himself.”