The Lions are reported to have offered the Ukrainian international a deal worth £1.65 million per-season.
Stoke City are also reported to be interested in signing the Ukrainian international midfielder according to the BBC.
The Istanbul gains plan to strengthen their squad ahead of the next league campaign and the Champions League group stages.
The Lions booked a place in the group stages after winning the league for a record 20th time last season.
Galatasaray will become the first Turkish side to wear four stars on their shirt next season. The Lions also won the double after lifting the Turkish Cup for the 16th times.
Konoplyanka reached the Europa League final with Dnipro last season but his side lost 3-2 against Sevilla.
The Ukrainian international is available on a free transfer after his contract ran out at the end of the season.