Mixu Paatelainen succeeded Jackie McNamara but United are 13 points adrift at the foot of the table.
Thompson apologised to supporters for "abysmal" performances.
"We have failed miserably on the pitch with only three wins in 25 league matches and an early exit from the League Cup," said Thompson.
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"The current position [in the league] is wholly unacceptable."
United fell to a 3-0 home defeat by Motherwell at Tannadice on Tuesday and have 13 league games remaining this season.
Paatelainen took up the managerial reins in October  and described as "wonderful" returning to manage a club he starred for as a player.
Thompson, in a statement on the club's website, challenged the playing squad to preserve their top-flight status and restore their reputations.
"It is not where Dundee United should be under any circumstance," the chairman added.
"The board and myself have made certain decisions in the last 18 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."
Speaking following Tuesday's defeat by Motherwell, Paatelainen said: "We were very flat and that performance was not acceptable.
"The first two goals were set-pieces and it's cost us. It's been the story of the season conceding goals like that and it's simply not good enough."
Stephen Thompson says Dundee United's squad have "13 matches to preserve the club's Premiership status and redeem their own professional reputations".