Ibrox Park, the home of the League One champions, was chosen to stage both semi-finals last October with the final being played at Celtic Park.
But after Rangers secured their place in the last-four with a replay victory over Albion Rovers on Monday, Thompson called for the match to be moved away from their home ground.
Thompson was also unhappy with his side’s allocation of 13,100 tickets and claimed the SFA were being ‘economical with the truth’ over the amount United has requested for the game.
He has now drawn a line under the spat insisting any future discussions with the governing body will be kept ‘in house’.
In a statement the Tannadice club said: “Regarding the William Hill Scottish Cup semi-final ticketing and venue issue, the club has held constructive discussions with senior officials from the Scottish FA and both parties have agreed to conduct future consideration on this matter in private.”
Meanwhile, Rangers manager Ally McCoist agrees his side could have an advantage playing the match on their home ground, but insists he would have taken any venue in the country had he been offered a last-four spot at the start of the campaign.
“It would be wrong of me to say it doesn’t really matter if we are the home team and it’s at our venue. I totally understand some of the points but the decision has been made.
“I do understand we are playing a semi-final at Ibrox and that is not to everyone’s cup of tea, but the decision was made by the member clubs that the semi-finals would be held at our ground and it will only prove an advantage to us if we win.
“At the start of the season we would have played a semi-final anywhere and we are just delighted to be there.”
Dundee United boss Jackie McNamara, whose side beat Rangers at Ibrox in the same competition last season, has backed his counterpart’s stance.
“It would be the same for us if the game was at Tannadice,” he said.
“So there is no point in me going on about it. We will just go down there and try and get through it.”
Rangers will play Dundee United on Saturday, April 12 with the other semi-final between Aberdeen and St Johnstone at the same venue the following day.