The 26-year-old, who returned to Dortmund from Bayern Munich in 2016, has made just two first-team appearances for the club this season.
"I'm very sorry for Mario Gotze at the moment, because stuff is written about him - and only him - every day," Grosskreutz told Sky in Germany.
"In Brugge, for example, the whole team didn't play well.
"Maybe it would be good for him to go abroad in the winter to find a bit more peace and bring his career back on track. Klopp knows him quite well, he would be a good fit for Liverpool."