The 26 year old midfielder has been in talks with Hearts this week and is expected to complete a deal sometime today.
The Boro born player, who has been at the club for over 9 years, has made over 100 appearences for the Teessiders, but starts over the last couple of years have been rare due to injury problems and lack of form. Last season he featured only 9 times, while he has yet to play this term.
Hearts will be hoping Stamp will help tighten up the midfield in there bid to qualify for Europe.