The Englishman is a free agent after leaving Preston North End at the end of the season following the expiry of his contract.
Currently a free agent, the 33-year-old is able to sign for a club outside of the transfer window.
On Instagram he hinted that the next move could be very soon with a post of him in Chelsea gear training accompanied by the caption: “Getting ready for my next move… [hourglass emoji]”
Sinclair has been linked with a return to Bristol Rovers where the forward began his career before Chelsea paid a six-figure fee to sign him as a teenager. His journey to Celtic was an interesting one, involving numerous loan moves, impressing at Swansea City and being signed by Manchester City for £8million.
He was a key recruit for Brendan Rodgers and would score 62 goals in 167 games for Celtic, helping the club win the treble treble.
In Sinclair's first season at Celtic Park, he produced a devastating campaign with 25 goals in 50 games. Across his time at the club he formed a fearful partnership with Kieran Tierney on the left flank.
He left in January 2020 to join Preston where he has spent the last two-and-a-half years.
If he was to join Rovers, he would be managed by Rangers flop Joey Barton. The club won promotion from League Two last season and have former Ibrox midfielder Jordan Rossiter and ex-Hibs loanee Sylvester Jasper in their squad.