The report claims that Reds boss Jurgen Klopp is keeping close tabs on the Germany international.
Klopp is of course no stranger to Grosskreutz having coached the hard working midfielder during his time at Borussia Dortmund.
Galatasaray signed Grosskreutz from Dortmund over the summer transfer window but were unable to register the 26-year-old due to failing to send over the necessary paperwork to FIFA before the transfer deadline.
Grosskreutz has been unable to play a single game for his new club as a result of the mishap and is reported to be unhappy with the situation.
Galatasaray manager Mustafa Denizli has claimed that Grosskreutz wants to leave the Istanbul giants due to the way his transfer was handled.
The versatile midfielder can play in many different positions but does have another three-years left on his contract.