Ms Soubry, one of 11 Conservative MPs to rebel against Theresa May on Brexit, said it was “our decision to walk away” and the threats are a “consequence of our decision”.
Her comments provoked an angry response from Hartley-Brewer, who snapped back at the Remainer over the EU’s threats.
The European Commission’s chief Brexit negotiator Michel Barnier levelled a series of threats against Britain during a press conference in Brussels last week.
Mr Barnier warned a transitional period for the UK government “is not a given” following a number of “substantial” disagreements between the two parties.
The Frenchman said: “If these disagreements persist, the transition is not a given.
“To be frank, I am surprised by these disagreements. The positions of the EU are very logical.”
Describing the threats, Ms Soubry used the analogy of Britain leaving a sports club while still wanting to use all the facilities, insisting the EU’s actions are somewhat justified.
The Conservative MP for Broxtowe told talkRADIO: “If you’re a member of a club and you said to that club, ‘I don’t want to be a member anymore, i’m going but would like to tennis courts on Monday, Tuesday and Wednesday… And, can aunty Flo use them on a Saturday…’”
Hartley-Brewer snapped back: “That’s not is what is happening!
“What is happening is they are saying, not only, we don’t want you to use our stuff, and that’s fine, you’re not paying for it so you don’t get to use it, but they now want to punish us!
“Even you, as someone who would like us to remain in the EU, even you Anna, you must be uncomfortable with some of the stuff the EU has said recently?”
Ms Soubry argued it was Britain’s decision to quit the European Union and it was a “big mistake” to think the threats are not fair.
She said: “The big mistake everybody is making is this is our decision, we’re not being thrown out of this club.
“We’ve said we want to leave your club, that is a big distinction. We’re saying we want to leave your club, but sort of, actually, we want to rejoin it.
“That is the point, we’ve decided to walk away… it’s our decision to walk away, we haven’t be slung out. These are the consequences of our decisions.”
Hartley-Brewer concluded: “The consequences don’t have to involve being threatened with punishment sanctions.”