The 24-year-old left the Swans for the Etihad Stadium last summer but the transfer has not worked out, with the former Chelsea man making just 15 appearances for Roberto Mancini's side, including only three starts.
The City boss has already admitted that it would be best for Sinclair to leave at the end of the season, prompting rumours of a return to the Liberty Stadium, and Britton is hopeful that the transfer can happen.
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"When you get an offer to go the champions you understand why he went and maybe it hasn't worked out how he wanted it to be.
"If he did become available - and obviously it is down to the manager - the players would welcome him back tomorrow."
The attacker missed his chance to play once more at the Liberty Stadium with City on Saturday as a blood clot in his shoulder left him in hospital.
Of Sinclair's illness, Manchester City captain Vincent Kompany said after the game: "We have all sent Scott our support. The news we got is that everything went well. We are very happy. We were all very scared when we found out he wasn't feeling well."