United announced less than two weeks ago that Houston would be leaving at the end of the season after protracted contract talks eventually broke down.
Houston stressed that he was focused on securing the fourth consecutive top-six finish of his reign and United took four points out of six from the subsequent two games.
But Saturday's 1-1 draw against Ross County was United's 11th home game in a row without victory and the speculation over Houston's successor appears to have created more problems for Thompson than the previous uncertainty over the future of the former Scotland assistant manager.
Houston departed little more than 24 hours before tonight's Clydesdale Bank Premier League clash with Motherwell at Fir Park and ahead of the keenly-anticipated William Hill Scottish Cup visit of Rangers on Saturday, although Thompson hopes to make an appointment before then.
Thompson said: "It's not been the best the last few weeks. I know in the continent it works very well when the coach says he's stepping down before the end of the season.
"It's maybe different in the UK, we're not quite used to that kind of thing.
"We've decided to move on and that's where we are."
Paul Hegarty takes charge of the team on Monday night with several names in the frame for the permanent job.
United are believed to have spoken to Falkirk manager Steven Pressley and Partick Thistle boss Jackie McNamara while former United player and coach Billy Dodds has confirmed he held talks with the club.
- Sports & Recreation