Miguel Herrera has been fired as Mexico coach following allegations that he punched a television reporter.
Former Arsenal midfielder Abou Diaby has rejected an offer to join West Brom in favour of finding a new club in France.
The former Holland defender's current deal expires at the end of the season but technical director Martin van Geel is hoping his stay at De Kuip can be extended.
Koeman guided Feyenoord to a second-placed finish last season and they are currently fifth in the Eredivisie, nine points behind leaders PSV Eindhoven.
Van Geel told Voetbal International: "We would love to (extend his deal).
"That we were second last year and now, even though we had to let some important players go, are fifth is clearly largely due to Ronald Koeman.
"We'll talk during the winter.
"We are very happy and I hope Ronald is too.
"As a coach, he has the necessary experience of highs and lows, and perhaps he cherishes the situation in which he can work."