At 5/1 shots, the Hoops have risen to second favourites with one bookmaker to sign the 37-year-old, who last played with Aston Villa, while Rangers and a host of American clubs are priced at 8/1.
Sky Bet has Derby County as 1/2 favourites to secure the veteran centre back’s signature but Celtic - who could be on the lookout for a new central defender if Dedryck Boyata or Jozo Simunovic leave - are also a mooted destination.
Terry spent 22 years at Stamford Bridge, where he won five Premier League titles, five FA Cups, three League Cups, one Europa League and one Champions League, but joined Villa on the expiration of his Blues deal in the summer of 2017.
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He led Villa to the Championship play-off final, where they lost out to Fulham, and the club’s financial state meant they were unable to offer Terry new terms at Villa Park.
The 78-cap England international was linked with Derby earlier this summer, after his friend and former team mate Frank Lampard took charge at Pride Park.
However, the Rams have recently tied up Curtis Davies on a new contract so their interest in Terry may have come to an end.
Terry was linked with Rangers back in May, when former international team mate Steven Gerrard took the reins at Ibrox, while in 2006 he revealed his backing for Celtic as a child.
Speaking after Chelsea and Celtic met in a friendly match, Terry said: “I have always been a bit of a Celtic fan. Like my brother Paul, I followed Scottish football as well as English football when I was young.
“He supported Rangers and I wanted some atmosphere in the house so I went for Celtic.”