The Serie A champions, who are on the verge of acquiring Benedict Howedes from Schalke 04, are expected to part ways with the Ghana international, upon the signing of the Germany defender.
Goal understands Galatasaray, who have already reached a principle agreement with Asamoah, have tabled an offer of €4 m to Juventus, although the Old Lady currently price the utility man at €7 m.
Faced with the situation of losing the Ghanaian on a free transfer next summer when his current deal expires, Juve may be forced to soften their stance.
The two clubs are expected to hold talks on Tuesday and should an agreement be reached, the 28-year-old could sign a four-year deal worth €2.5 million a year.
An imminent move away became apparent after the Accra-born was left out of Saturday’s squad for the league clash with Genoa, with coach Massimiliano Allegri confirming the Ghanaian is “evaluating his future” at the Allianz Stadium.
Asamoah has been reduced to a squad player at Juve over the last three seasons, and injury problems have done little to help his cause.
He has made 36 league appearances in all since summer 2014, having transferred from Udinese in 2012.
Five Serie A titles, three Coppa Italia winner's medals and three Super Cup golds make up the Accra-born’s trophy haul.