The Black Cats are staring at the possibility of a second successive relegation, given their current position at the bottom of the Championship table after 16 games.
Coleman was appointed Wales manager in extremely hard circumstances in January 2012 following the death of his great friend Gary Speed.
The FAW have also confirmed Coleman's assistant Kit Symons' resignation. "The FAW and Wales as a nation will be eternally grateful for the job he has done over the last six years as National Team Manager", FAW chief executive Jonathan Ford said in a statement.
Reports on Tuesday morning hinted that Sunderland wanted to get a deal with Heckingbottom over the line in time for their home clash with Millwall on Saturday.
"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".
Trump and his awkward way of "staying hydrated"
Tables have now turned for Rubio, who tweeted right after the video went viral: "Similar, but needs work on his form". Seeing Trump in the awkward position, Rubio did not miss a chance to take a jibe on the US President.
Former Fulham and Coventry City boss Coleman will team up with Wales internationals Jonny Williams, on loan from Crystal Palace, and right-back Adam Matthews at Sunderland - both of which have played under Coleman at global level.
Disappointment followed as Wales failed to reach the 2018 World Cup finals in Russian Federation.
He said: "I hope that's not going to be the last game".
"I actually feel the north east for football in particular is one of the places in the United Kingdom but I sense the Sunderland fans for the first time I can remember have lost a little bit of heart and a little bit of belief".
Michael O'Neill, the Northern Ireland manager, Paul Heckingbottom, who is in charge at Barnsley, and Ally McCoist were among the other candidates considered for the Sunderland post but Coleman emerged as the clear frontrunner.