The Teessiders hope to conclude a deal for the 24-year-old Dubliner within the next 48 hours.
Yeates, who it is understood would cost in the region of £500,000, has been at the club's Rockliffe Park training headquarters for a medical and to discuss personal terms.
His capture would cap a positive week for manager Gareth Southgate, who has already managed to persuade utility man Matthew Bates to sign a new three-year contract.
Southgate told the club's official website: "We have been in negotiations with Colchester and we feel we are close to being able to do some sort of a deal there.
"Hopefully something might happen over the next couple of days."
Colchester confirmed that a deal is close on their own website.
Chairman Robbie Cowling said: "We have a history of trying to keep our best players, but it is important to recognise when the time is right for all parties to allow them to leave the club.
"We have received an offer we feel is acceptable and assuming that Mark and Middlesbrough agree everything on a personal level, we feel we are giving him the chance to continue his career elsewhere."
Former Tottenham trainee Yeates scored 12 times in 42 appearances for the League One club last season.
His prospective arrival comes as Southgate attempts to re-shape his squad for life in the Championship amid interest in both Stewart Downing and Adam Johnson.
Downing, who is recovering from foot surgery which could sideline him until well into the new season, has once again been linked with Tottenham among others, while natural replacement Johnson too has his suitors.
But whatever happens on that front, the manager is confident Yeates has all the attributes he is looking for in his summer targets.
Southgate said: "We have got to be mindful of covering everything.
"We need players of Premier League quality, but I want players who have played in the Championship and who have a hunger to push on and a desperate desire to play in the Premier League.
"We want a mixture of hunger, energy and quality."