Terim has been out of work since July, when he left his role as Turkey's national team coach after he was involved in an altercation with a restaurant owner.
Gala sacked Igor Tudor on Monday after it lost three of its last four matches in all competitions under the former Juventus and Croatia defender.
Terim has not worked in club football since 2013, when he left Galatasaray to re-join Turkey after claiming his sixth Super Lig title, all of which have come at the Istanbul giant.
The 64-year-old - who also represented the club as a player - has a long association with Galatasaray, having first taken charge in 1996 after leading Turkey to the European Championship in the same year.
Terim subsequently won four consecutive Super Lig titles to write his name into Gala folklore, also claiming the 2000 UEFA Cup crown, although he did not succeed during short spells in Italy with Fiorentina and AC Milan.
During Terim's second spell in charge of Turkey, he led his country to the semi-finals of the 2008 European Championship.
Galatasaray sits third in the Super Lig table after 16 matches, a point behind leader Istanbul Basaksehir but ahead of fourth-placed champion Besiktas.