Sinclair has started only two Premier League games for City since he joined from Swansea City last summer and has been linked with a loan move away from the Etihad Stadium.
Everton and Newcastle United are reported to have shown an interest in Sinclair, but Clarke is hoping to win the chase for the 24-year-old.
Clarke has so far only made three additions to his squad this summer and is keen for further arrivals before West Brom start the new season at home to Southampton next Saturday.
"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."
Another player West Brom have targeted is Kalou after the Lille attacker's £2.6million release clause in his contract was triggered.
Kalou has also been linked with West Ham United, but Clarke does not expect the former Chelsea player to leave Lille before the transfer window closes.
Clarke added: "I don't think anybody has been able to persuade Salomon to come out of France.
"The closer you get to the start of our season, the closer you get to the start of the French season. It doesn't look as if that one will happen, but that's normal in any transfer season.
"It's not just us who couldn't persuade him. A few clubs wanted him."