The 24-year-old left the club last summer, joining the Citizens in a deal worth £6.2m, but has recently been linked with a move back to the Liberty Stadium after struggling to break into Roberto Mancini's first team.
"We'd be happy to have him back," Britton told The Mirror. "Scott did great in his two seasons here. In the Championship season he scored 25 goals and was a massive part in the success we've had.
"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, but the players would welcome him back tomorrow."
Sinclair was taken to hospital on Friday with a blood clot in his shoulder.