United are charging fans half price for the December 15 Clydesdale Bank Premier League game, which came about following the abandonment of the sides' November 1 meeting at Tannadice.
Thompson has hit back, telling BBC Scotland: "I think Martin's behaviour has been disgraceful."
The original fixture, which Rangers were leading 1-0, was halted at half-time due to a waterlogged pitch.
Thompson insists fans were never told their original ticket stubs would be valid for the replay and reduced pricing for the rescheduled match was a fair reflection of the operational costs incurred by United.
The two clubs have been rowing over the matter ever since, with Bain claiming on Monday: "The game was called off at half-time and I was standing beside the chief executive of the Scottish Premier League (Neil Doncaster) and the chairman of Dundee United, and I said to them at that time, 'I hope that you will see sense and make sure that our supporters retain their ticket stubs and get back in for nothing'."
However, Thompson denies Bain ever said this in his company.
"It is interesting that Martin Bain was disgusted and annoyed with Dundee United over tickets but never used such strong language after Manchester or Bucharest about his own fans," added the United chairman in reference to recent crowd disturbances involving Rangers supporters.
Bain has now told his club's official website: "After being accused that I used the AGM as a platform to criticise Dundee United and cover up the club's issues, the reality is that the volume of complaints and the tremendous ill feeling that was being vented towards Dundee United's decision was a matter that simply could not be ignored.
"It is very disappointing as our position could not have been made clearer at every stage since the game was called off.
"There is obviously written correspondence between the clubs that confirms this.
"I deeply resent the suggestion that I misled shareholders at our AGM. That is wrong and offensive and I will be taking the matter further.
"I still maintain that football fans have been considered last by this decision."
"What they are now suggesting is completely out of order and the Rangers supporters know the truth of what happened here.
"Dundee United has behaved very badly throughout this whole saga."