The veteran former England international spent last season at Everton, where he was a first-team mainstay as the Toffees narrowly missed out on a shock top four finish.
Roberto Martinez is keen to keep the 33-year-old at Goodison Park, but even though he is set to be a free agent in the coming months they would have to pay £2.5m due to a clause inserted into the deal that took him to Merseyside.
Although he has been a key player, Everton are reluctant to shell out such a figure for Barry due to his age.
This has put Tottenham in pole position, with the club able to snap him up for free.
Spurs are rumoured to be looking for a fresh face to add to their midfield and chairman Daniel Levy sees the experienced former Aston Villa man as a riskless addition.
Barry himself is likely to be open to the move, with the Lilywhites expected to mount a charge for European qualification next season