The English winger’s contract at Park Head is due to expire at the end of this season and the Hoops have reportedly set a release clause of £1.75m.
The 30-year-old was signed from Aston Villa in 2016 in a sumptuous deal.
He has made 150 appearances since and scored 50 goals for the Scottish kings.
Furthermore, he was instrumental when the Bhoys achieved a domestic treble last season.
However, the former Swansea man became out of sorts this season, where his goalscoring stats were less than that of previous seasons.
Sinclair stated his uncertainty regarding his future in Scotland earlier this year.
Now, the Boros, who are currently in a promotion bid, are likely to trigger the release clause of the Hoops forward.
Sinclair’s exit- whether a blow for the Gers
Sinclair has not been at his best this season, scoring 10 goals in more than 40 matches across all competitions.
Meanwhile, newly bought forwards like Timothy Weah and Oliver Burke also failed to live up to their expectations.
The Celts were highly reliant on their go-to guy James Forrest and Odsonne Edouard in the title run this season.
The Scottish club now can highly benefit from Sinclair’s deal and then they can invest in promising talents.
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Neil Lennon, the new caretaker manager, is likely to continue as the next Hoops manager on a permanent basis after achieving seven wins and a defeat.
The Bhoys will be facing Hearts in the Scottish Cup final this month.