The alleged incident took place immediately after the two clubs met in the Betfred Cup match at Dens Park on Wednesday.
It is understood Mr Thompson, 51, was allegedly pushed to the ground during the incident, but was not injured.
Dundee won the game 2-1 and will meet Celtic in the quarter-final.
A Dundee United spokesman said: "We can confirm there was an alleged assault on chairman Stephen Thompson immediately after the final whistle at Dens last night.
"While shocked and upset at the incident, the chairman did not require any medical treatment.
"He remains deeply disappointed that his wife and son had to witness the incident."
A 24-year-old man has been charged with assaulting Dundee United chairman Stephen Thompson following the club's defeat by city rivals Dundee.