The 24-year-old, who has previously been linked with Liverpool, was a standout performer for the German outfit in the 2021-22 campaign, scoring three goals and registering 13 assists in 32 Bundesliga appearances.
According to 90min, Raum is expected to leave Hoffenheim this summer and the Citizens have made contact to discuss the player's availability.
The report adds that Hoffenheim are interested in signing City's Japanese defender Ko Itakura, who spent last season on loan with Schalke 04 and who is yet to make a single first-team appearance for the Citizens since joining the club in 2019.
A host of other clubs, including the likes of Celtic and Borussia Monchengladbach, are also believed to be interested in Itakura, but Hoffenheim could be willing to negotiate a swap deal with City which would involve Raum heading to the Etihad Stadium.
The report also states that Pep Guardiola's side are still prioritising a deal for Brighton & Hove Albion's Marc Cucurella, with City allegedly poised to make a bid for the left-back, but they have enquired about Raum in case they fail to lure the Spaniard away from the South Coast.
Raum, who has four years remaining on his Hoffenheim contract, has made nine international appearances for Germany and featured in all four UEFA Nations League matches earlier this month.