The report states that the Germany international would like to join his former manager Jurgen Klopp at the Premier League side.
Galatasaray signed Grosskreutz from Dortmund last summer but were unable to register the 26-year-old after failing to send over the necessary paperwork to FIFA before the transfer deadline.
Grosskreutz will not be able to play for the Lions until January when the transfer window opens.
Liverpool were reported to have agreed personal terms with Grosskreutz according to the Turkish newspaper Takvim.
Scottish giants Celtic have also joined the race for the World Cup winner according to the Scottsman.
The report claims that the the Lions are happy to let the former Borussia Dortmund man leave for a fee in the region of £1.8 million.