The report claims that the latest bid from Brendan Rodgers' side stood at £2.75million plus add-ons but Villa value their player at closer to £5million.
The Hoops have previously made unsuccessful bids of £1m, £1.25m and £2.2m for the 27-year-old, who still has three years left on his deal with Villa.
Celtic boss Brendan Rodgers is determined to lure Sinclair to Parkhead having struck up a successful partnership with the former Chelsea and Manchester City man at Swansea City.
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