The Scottish Premiership side have reportedly wrapped up a deal to sign the 27-year-old.
The Mail claims that Sinclair has completed his move to Parkhead having signed a four-year deal with the Hoops. And while he is arguably under more pressure than either of the other two signings that Brendan Rodgers has made this summer to hit the ground running - with Moussa Dembele still in the early stages of his career, while Kolo Toure is in the twilight of his - there is perhaps an acceptance that the 27-year-old is yet to realise the potential he showed when he burst onto the scene.
But McLeish - who was Birmingham City manager when Sinclair went on loan to St Andrew's in 2009 - is confident that joining the Bhoys will allow the winger to prove that he can be as good as everyone expected him to be earlier in his career.
"In terms of the title and winning trophies, signing Scott is a big statement from Celtic," he told the Mail.
"He'll be anxious to ressurect his career again and show why he was a young protege. He was feted by everybody as a kid.
Sinclair's spell with Swansea City was arguably the best of his career
"He really is the real deal but he's still got that challenge ahead of him at Celtic. And he's very capable of doing it."
While there are perhaps several reasons why Sinclair has struggled to make the impact he threatened to make as a youngster, a key moment in his career appears to be his decision to move from Swansea to Manchester City in 2012.
Celtic manager Brendan Rodgers
Having been a talisman for the Swans, he made the decision to join a club more likely to be battling for silverware and competing in the Champions League. And that switch proved costly as his career appeared to stagnate for a while.
This move to Celtic has arguably seen him make a similar decision, but with competition less fierce within the Hoops' squad and with a manager at the helm who got the best out of him previously, Sinclair now has the opportunity to establish himself as a talisman in a successful team.