Sinclair has failed to make the expected impact at Eastlands since joining City from Swansea City last summer and could be set to leave the club.
Everton and Newcastle United have also been linked with the 24-year-old winger.
"We've made an enquiry to Manchester City," said Clarke in the Birmingham Mail. "There are discussions between the two clubs but beyond that there is nothing to report.
"It would be nice to think we can get one in before Saturday. If we can get two in, it would be great. If I can get three, I would be delighted.
"I know we have until September 2 - at the moment I must confess I'm a bit anxious, but I'm not panicking yet.
"I'm the head coach. I give them (the club) a list of players who I think would be good for us. By and large the targets we've identified we haven't been able to persuade to come so far. Some of them have maybe been too expensive."