The German World Cup winner is known to be keen on a return to the Bundesliga.
The report in Hannoversche Allgemeine Zeitung claims that Hannover have been in touch with Galatasaray about the player, who would prefer to continue his career in Germany this season after his Turkish nightmare.
The report goes on to state that the club are willing to pay the €3 million (£2.2 million) asking price for the player, although much of that will be made up of a player exchange. Turkish striker Mevlut Erdinc has struggled in German football and the club are willing to offer him to the Istanbul club as a makeweight.
Turkish paper Takvim have reported this week that Grosskreutz, who won the World Cup with Germany in 2014, had agreed a three-year-deal with Liverpool.
Grosskreutz has not featured once for the Turkish side since leaving Borussia Dortmund in the summer and is now said to be up for sale. He was not registered in time by Galatasaray for the first-half of the season and he has become disillusioned with the situation.
He has however believed to be keen on a return to Germany and Hannover would therefore appear to be the new favourites for the player.