The report claims that the Istanbul giants are happy to let the Germany international leave for a fee in the region of £1.8 million.
The daily paper stated that the Premier League side have agreed a three-year deal with the Germany international but an agreement with Galatasaray is yet to be reached.
The Reds or Galatasaray are yet to confirm any deal being in place although head coach Mustafa Denizli did admit that Grosskreutz has expressed a desire to leave.
The Istanbul giants 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.