The Terrors, who are currently at the bottom of the Scottish Premiership table, suffered a 3-0 defeat at Tannadice Park.
On Wednesday, Thompson issued a statement, which read: "As chairman of the club, I would like apologise to all Dundee United supporters for the abysmal performance last night against Motherwell. The current position at the bottom of the Ladbrokes Premiership is wholly unacceptable. It is not where Dundee United should be under any circumstance.
"The board and myself have made certain decisions in the last eighteen months which in hindsight were incorrect but were made for the greater good of the club and with the ambition of maintaining our position at the top of Scottish football, something we have done successfully for the last eight years. This included supporting a playing squad that commands the third highest wage bill in the Ladbrokes Premiership.
"However, we have failed miserably on the pitch with only three wins in 25 league matches and an early exit from the League Cup. Performances such as last night simply cannot be tolerated. The current playing squad now have 13 matches to preserve the club's premiership status and redeem their own professional reputations."
The team have earned just 14 points from 25 games.