The Spurs midfielder seems unlikely to stay at Tottenham for the whole 2016-17 season, with Mauricio Pochettino boasting an array of stronger midfield options.
And the 24-year-old, who started four Premier League games last season, is 10/1 to join up with David Moyes at Sunderland, according to prices by Bet Victor.
But Swansea City and West Bromwich Albion are both in front of the Black Cats on the list, with both Premier League outfits quoted as 6/1 co-favourites to sign him.
Carroll spent a season on loan at Swansea - then under Garry Monk, now under Francesco Guidolin - during the 2014-15 campaign, but managed to start only eight league games during his time on South Wales.
Sunderland manager David Moyes before the match
West Brom have only spent £5.5 million this summer on Matt Phillips who remains their only signing of the transfer window. The Baggies could do with reinforcements after selling or releasing five players.
There is some confusion regarding Carroll's Tottenham contract. The academy graduate was officially out of contract with Spurs in June but he remains on their books, though the club have not announced an extension.
West Ham, Brentford and QPR - where Carroll spent the 2013-14 season - are 12/1 to bring him in this month.