The sort after Ukrainian international has also been closely linked with Tottenham, Stoke and Europa League champions Sevilla.
The Lions are reported to have offered Konoplyanka a deal worth £1.65 million plus bonuses per-season.
Galatasaray are planing to bolster their attacking options ahead of the next league campaign and the UEFA Champions League group stages.
The Istanbul Giants booked a place in the UEFA Champions League group stages after winning the league for a record 20th time last season.
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.
Konoplyanka provided seven assists and scored eight goals in 42 appearances for Dnipro in all competitions last season.