Sahin representatives have been in Madrid looking to arrange the 23-year-old's departure after it became clear he was no longer in Jose Mourinho's first-team plans.
Liverpool have also expressed an interest in the Turkey international, but Arsenal are reported to be his preferred destination, although that will be on a season-long loan rather than a permanent move.
Manager Arsene Wenger had made it clear he would still be open to more new arrivals, having already brought in Olivier Giroud, Lukas Podolski and Spain international Santi Cazorla, who impressed in midfield on his debut against Sunderland at the weekend.
"We are looking to bring one more midfielder in of course, and maybe one defender as well because we still feel we are short in some positions," Wenger told Arsenal Player.
Long-time target Yann M'Vila is still on the radar. However, Rennes have named Tottenham as one of the firm interested parties, with Zenit St Petersburg and another anonymous club also at the table.
The arrival of Cameroon defensive midfielder Jean Makoun on loan from Aston Villa could further hasten M'Vila's departure, although Rennes general manager Pierre Drerossi has not yet ruled out the player staying for another season if their £17 million price tag is not met.
While Wenger could have reinforcements in place for Sunday's trip to Stoke, it is clear the manager expects a much-improved attacking display after drawing 0-0 against the Black Cats despite dominating for long spells.
He said: "We didn't lose, but it is frustrating when you are home with everyone here and you do not create more chances than we did.
"It is very difficult to understand."