The Gunners are reportedly interested in signing Sahin on a season-long deal with an option to buy, but face competition from Galatasaray for the 23-year-old Turkey midfielder.
Emirates Stadium is Sahin's preferred destination and the transfer moved a step closer when Mourinho declared he is surplus to requirements at the Bernabeu.
"The point is that he's a young boy who wants to play every match," the Real manager said.
"He wants to play to be happy and in Real Madrid with Xabi Alonso, Sami Khedira, Lassana Diarra, Esteban Granero and Mesut Ozil, the situation is not easy.
"Real Madrid is very difficult. It is not an easy club, because we demand a lot. We have great players and when you don't start well, it's difficult to do it.
"If he stays, he's not a problem for me. He's one more solution. We are leaving the situation in his hands.
"Decide what you want to decide, to be happy. If you decide to leave we're going to help you to leave. If you decide to stay, we're going to support you and try to succeed this season."
Sahin joined Real for £8 million last summer after helping Borussia Dortmund win the Bundesliga title.
Injury prevented him from making his debut for the Spanish champions until November and he only made 10 appearances throughout the season, scoring once.
Arsenal's squad looks set to be strengthened by the arrival of Santi Cazorla, the Malaga attacking midfielder who is reported to have completed a medical. Cazorla has agreed a four-year contract and the highly-rated 27-year-old Spanish international is thought to have cost the Gunners £16 million.