But the Tynecastle manager does not believe the Tannadice chairman should even be considering axing manager Mixu Paatelainen despite the Taysiders' hopeless Ladbrokes Premiership predicament.
United look destined for relegation after losing 3-0 to rivals Motherwell on Tuesday.
It was a huge blow for Paatelainen and his team but it left Thompson furious.
The Tangerines chief issued an explosive statement the morning after the defeat in which he dubbed the team's performance "abysmal".
He also appeared to hint that hiring Paatelainen as Jackie McNamara's successor last October had been one several "incorrect" board decisions made over the last 18 months.
Neilson has sympathy for his opposite number - who he faces at Tannadice on Saturday afternoon - but defended Thompson's right to voice an opinion.
He told Press Association Sport: "I didn't read the statement but every chairman has the right to say what they want about their team
At the end of the day, they are the ones who put the money in.
"If he wants to say it that's up to him
Whether it gets a positive result or negative one, only time will tell.
"He obviously thinks it will provoke a response or he wouldn't have done it."
Paatelainen's position is reported to be safe for now but with the team 13 points adrift at the bottom of the table there is already mounting speculation that the Finn's days in the City of Discovery are numbered.
Neilson, however, believes the former Kilmarnock boss should be given time.
"I don't think Mixu is under pressure," he insisted
"He took over the club when they were bottom of the league.
"I would like to think he was brought in to stabilise the club first and foremost but also be given the time to build it.
"What can you do in four or five months? You get one window to work with when he didn't have a lot of money to spend.
"He spent the majority of the window trying to move people on and balance books
It's very difficult for a manager to do that.
"So I don't see Mixu being under any pressure
I think he'll get this season and future seasons as well because he's got a reputation of being able to build teams like he did with Kilmarnock and when he did fantastically well with the Finnish national team."
Neilson, though, will hope his team can pile more misery on top of United as they look to bounce back from Tuesday's William Hill Scottish Cup exit to city rivals Hibs.
They return to league duty on Tayside and Neilson is already eyeing a race for second place with Aberdeen.
He said: "It's always frustrating when you lose a derby but we have to move on, accept the criticism and try to build from here.
"We focus again on the league now
We've got two games in hand on Aberdeen
If we win those we go five points behind and then we can start looking to get after them
We still have a game against Aberdeen so they can be caught.
"That is the motivation for the rest of the season
We believe with the players we have got we can do it."
Source : PA