The 2014 World Cup winner returned to his roots at the Westfalenstdion in 2016 after spending three years with Bayern Munich.
Big things were expected of the 26-year-old star once he was back in familiar surroundings, but a homecoming has become something of a disaster.
Gotze has struggled for form and fitness since re-signing for Dortmund, with a midweek Champions League outing against Club Brugge representing his first start of the 2018-19 campaign.
Former club and international team-mate Grosskreutz feels a fading presence needs a switch to reignite his spark, with Liverpool a logical step due to the presence of Jurgen Klopp.
He told Sky Deutschland: “I’m very sorry for Mario Gotze at the moment, because stuff is written about him – and only him – every day.
“In Bruges, 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.”
Grosskreutz, who tasted World Cup glory alongside Gotze four years ago, added: “Of course he has to deliver performances on the pitch. His head is not completely free at the moment.
“I never expected his career to experience such a phase.”
Grosskreutz is not the first to have advised Gotze to look elsewhere for regular game time, with former Germany captain Lothar Matthaus having done likewise.
"His form, which he had before the 2014 World Cup and included his winning goal in the final, is behind him,” Matthaus said.
"For him, you would have to invent a system at Borussia Dortmund that does not exist right now.”