Arsenal have been linked with Arthur throughout the summer, but City could be set to beat the Gunners to Arthur's signature.
According to Calciomercato, via the Mirror, the Premier League champions are considering a deal with Juventus that would result in Gundogan and Arthur swapping clubs.
Gundogan's playing time could decrease this season following the arrival of Kalvin Phillips, and with his contract set to expire next summer, City are looking to avoid a situation where the Germany international leaves on a free transfer.
Arthur, meanwhile, was previously involved in a swap deal with Miralem Pjanic that saw him depart Barcelona for Juventus, but he is set to leave Turin this summer after making only 11 Serie A starts last season.
City could be set to beat Arsenal in the race for the Brazil international if they are prepared to allow Gundogan to move in the opposite direction.