Houston revealed that he would not be renewing his contract when it expired at the end of the season but said he see out the remainder of the campaign. Reports that Falkirk manager Steven Pressley and Partick boss Jackie McNamara had been interviewed for the job was followed by the announcement that Houston had left United by "mutual consent".
Assistant boss Paul Hegarty will take charge of Tuesday's Clydesdale Bank Premier League game against Motherwell but a new manager could be appointed before the visit of Rangers. And Thompson said: "You'll find out in the next couple of days where we are. I would prefer someone in place before this weekend."
He added: "We've been working on it for the last wee while, quietly behind the scenes. We've had a very long list and we've taken our time.
"We've actually had time to look at what we would like and what we are looking for, which we have been doing for the last 10 or 12 days to be honest. Now it's time for us to move on from there."
Pressley confirmed an approach from United before another former United player, Billy Dodds, revealed at the weekend that he had held talks with the club.
Former captain Derek McInnes was initially one of the favourites for the post, having been sacked by Bristol City earlier this month, but it is understood he has not been interviewed.
Thompson refused to be drawn on any names reported to be in the running for the position, adding: "I'm not going to confirm who I've spoken to and who I haven't spoken to."
Asked why the decision was taken to bring forward Houston's departure, the Tannadice chief said: "It's something I had been thinking about for a few days and that's the decision that has been made. A lot of thinking has been going on.
"As a club, and as a board, we decided we wanted to move on from the position we have been in. We do wish him all the best for the future. He's done a great job for the last two or three years, as well as the previous three years. It's just time for both parties to move on."