The loss of the former England Under-21 international seems likely to follow the blow of Joe Allen's departure to Liverpool for £15 million earlier in the transfer window, with Goal.com understanding that the Team GB Olympic star has agreed terms with Manchester City.
Swans chairman Huw Jenkins and manager Laudrup had hoped to convince Sinclair to extend his stay in south Wales, but the Dane revealed the former Chelsea player does not want to discuss terms.
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The forward's swift exit now seems inevitable as Swansea will not want to risk losing him for nothing next summer, but Laudrup made it clear he will not hesitate to pick him for Saturday's Premier League opener at QPR.
"I have spoken to Scott and he confirmed what he said to the chairman, that he won't sign a new contract," Laudrup told reporters.
"But he is here now and that's what important for me. Until August 31 anything can happen.
"Of course there is interest from a big club in Scott, but the only thing I care about in the short term is that it does not affect him for Saturday.
"If it affects the player then we have to talk about. I will talk to Scott and if he says no, he does not want to play, then we will see. If he says, 'I'm a professional', then I know he is in the squad.
"I don't know how official it is but there is interest," Laudrup claimed of City's pursuit.
"Manchester City is the name that is always coming up but again I have not spoken to them."
Sinclair notched eight goals last season as Swansea finished 11th in the Premier League after scoring 27 times in the Championship before that, netting a hat-trick in the play-off final win over Reading at Wembley that took his side to the top flight.
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"If it's two or one it is okay, but if you are talking about seven into the first XI you will need time to adjust.
Laudrup is reportedly interested in bringing Valencia's Pablo Hernandez to the club and the former Mallorca boss admitted that he is a fan of the winger.
"I had Pablo in Getafe and we had a fantastic year together. He then signed a five-year contract with Valencia," said the Swans boss.
"He is a player that I know very well, and he knows me."