David Moyes wants to capture Kostic, who impressed in the Europa League last season, but there has been a question mark over whether the player would agree to the move.
West Ham have put in an offer for Kostic which, according to German magazine Kicker, is slightly higher than the €13m widely claimed.
And it is suggested that West Ham progressing to the official offer stage strongly hints that they already have an agreement in place with Kostic.
The left-sided player would see his salary skyrocket if he makes the move to West Ham, but he would give up the chance to play Champions League football next season.
Juventus are also keen on Kostic, but they have not put in a bid with Eintracht Frankfurt.
The German side rejected an offer from Lazio last year, which came in at €10m.
Kostic, a Serbia international, provided 15 assists in 42 appearances for Eintracht Frankfurt over the course of last season.
