Coleman will succeed Simon Grayson, who was sacked after 18 games in charge.
The new challenge facing Coleman will be to steer bottom-of-the-table Sunderland up the Championship and away from relegation danger, with his appointment expected to be confirmed by Sunday.
The Sunderland managerial race continues to rumble on at the moment with Ally McCoist the latest man to now confirm that he would be interested in the job.
He took in disappointing spells at Real Sociedad, Coventry and AEL before rebuilding his reputation on the worldwide scene with Wales. However, he was unable to build on the momentum from their successful France 2016 campaign and qualify the team for next summer's 2018 FIFA World Cup Russia™. Robbie Stockdale will remain in charge in a caretaker capacity for Saturday's home game against Millwall, with Coleman set to be in the dugout for the first time on Tuesday night, when Sunderland travel to Aston Villa.More news: Uber Faces Lawsuits from Female Passengers Who Allege Rape
"We wish Chris the very best of luck for the future as he returns to club management, a desire for which he has always been honest and open about".
Heckingbottom distanced himself from the role when quizzed about being linked to the job before the worldwide break.
Heckingbottom is understood to be in the running, while former Middlesbrough boss Aitor Karanka has been linked with the vacancy - although sources close to the Spaniard have distanced him from it - and Ally McCoist, who worked with Bain at Rangers, has also come under consideration.
Coleman spent six months in Spain before returning to England with Coventry but left two years later before a short spell with Greek side AEL.