The Brexiteer asked when the "drop-dead moment" to walk away would come.
Mr Paterson is considered influential, as a former minister who wants a hard Brexit.
British Prime Minister Theresa May has appointed Jeremy Hunt as the new Foreign Secretary, following Boris Johnson's resignation from the job amid deep divisions over Britain's departure from the European Union (EU) or Brexit.
Following Mrs May's Commons speech, Wrekin MP Mark Pritchard praised her "robust performance". I think she is strengthened by all of this.
Mrs May has said she wants talks to go ahead with greater intensity and pace once Britain has set its position out in the white paper due for publication later this week.
"I don't see them pulling the plug on her", Polly Mackenzie, director of cross-party think-tank Demos, told AFP.
Acting Prime Minister Winston Peters told Newsroom that May's announcement was a validation of the work that the Government had put into renegotiating controversial aspects of the CPTPP.
Le Figaro, the conservative French broadsheet, says the prime minister has been "destabilised by a pro-Brexit revolt", although the inside editorial points out the "mutiny is far from having been won".
"I think she (Mrs May) showed authority on Friday, has provided a lot more certainty in respect of the British approach towards the Brexit negotiations and I expect her to see that through now", Mr Coveney said.
In a tweet clearly created to show she was not being knocked off course, May said: "Productive Cabinet meeting this morning - looking ahead to a busy week".
"Any attempt by the European Union to whittle away at the current proposal would see a significant number of MPs conceding that no deal is better than a bad deal".
Iniesta calls time on Spain after World Cup exit
Spain have only themselves to blame for they never should have allowed this game to go into the Russian roulette of penalties. With 131 appearances, Iniesta was Spain's fourth-most-capped player, behind Iker Casillas, Sergio Ramos and Xavi Hernandez.
Earlier in the day, Mrs May was bolstered by the support of senior Brexiteers in her Cabinet.
Taking charge of the Foreign Office, Hunt said he would be standing "four square" behind the Prime Minister "so that we can get through an agreement with the European Union based on what was agreed by the Cabinet last week at Chequers".
Downing Street made clear the PM would fight any attempt to oust her by rebel MPs.
British Prime Minister Theresa May won support from many of her ministers, including a leading eurosceptic, on Tuesday after two top cabinet members quit saying her Brexit plan was too half-hearted.
Not since 1982 - over the Falklands, a literal war - have two Secretaries of State resigned within 24 hours. With genuine disappointment, I have to say I hope so. "But there are people in Labour and the Liberal Democrats who wish to damage the sort of Brexit that many of my constituents have voted for".
"I never thought I would say that again, but the government's sell-out leaves me with no choice". I have said that and so has the Taoiseach.
"I assume we'll continue to be promiscuous about these negotiations: we'll try everything with nearly everyone".
His resignation comes just a month after he was heard launching a scathing attack on the government's Brexit strategy in a leaked tape from a private dinner with Conservative activists.
However, she has support from moderates and there has been no formal challenge to her leadership so far.
It said May now faces "the spectre of a leadership challenge", but it was "possible that after a period of resignations and political blood-letting, the Conservative Party will fall behind the prime minister".
I've always liked him.' Trump and Boris clearly have an affinity which Theresa, bless her, can't compete with.
Each future Conservative leader must be able to perform their own party balancing act, and the few ambitious enough to seek the opportunity are unable to do so.