The Turkish sports newspaper claim that the 31-year-old wants to remain at the Emirates until the end of his contract.
Flamini only has a year left on his current deal and could end up leaving on a free transfer unless Arsenal offer an extension.
Eurosport TR claimed that Arsenal had accepted a £4 million bid from Galatasaray for Flamini and permitted the Lions to enter talks.
Flamini provided two assists and scored on goal in 33 appearances for Arsenal last season.
Galatasaray already have strong links with the North London club following the transfer of Lukas Podolski this season.
The Istanbul giants have been targeting a new defensive midfielder following reports Felipe Melo could be on his way out of Istanbul.
Galatasaray won the league title for a record 20th time last season and booked a place in the Champions League group stages for the fourth consecutive season.