The 28-year-old boyhood Hearts fans is entering the final six months of his contract and is not a first-team regular under current manager Robbie Neilson, making only four league appearances this season.
Fellow cinch Premiership clubs Livingston and St Johnstone had hoped to take Walker on loan, but English League Two outfit Bradford City – managed by ex-Ross County boss Derek Adams – are now in pole position to sign him.
Walker has previously played in England, spending 18 months at Wigan Athletic after moving there in 2018.
The Edinburgh Evening News reports that Hearts were hopeful of persuading Walker to stay, but were “sympathetic” to his desire for more regular first-team football.
Walker is now en route to Yorkshire, where he is expected to seal a move to the Bantams.