It comes after Mrs May told Tory MPs: "I'm the person who got us into this mess and I'm the one who will get us out of it".
After losing her parliamentary majority in a botched gamble on a snap election, Ms May is so weakened that her Brexit strategy has become the subject of public debate inside her own party with calls for her to take a more business-friendly approach.
"We have had some very good discussions again today. and I hope that we can reach a conclusion sooner rather than later", Ms Foster told Sky News.
The chaos has also weighed on the pound, which has plunged nearly two percent since Thursday, and the government may have to delay the announcement of its policy plans to parliament.
A potential agreement for the Northern Irish unionist DUP to prop up a minority Conservative government led by Theresa May in Britain is causing anxiety and fear in the province, Irish nationalist party Sinn Fein said on Tuesday.
Passing the Queen's Speech - which is written by ministers and presents an outline of its planned legislation for the next Parliamentary session - will be the first major test of Theresa May's proposed minority government.
However, George Osborne, a former finance minister who is now editor of London's Evening Standard newspaper and a vocal critic of May, said she appeared a "dead woman walking".
Since coming to power, May has advocated a "hard" Brexit, which would entail Britain leaving the European single market and cutting immigration from the bloc.
"We stand at a critical time with those Brexit negotiations starting only next week - I think that stability is important".
Delaying talks does not delay a Brexit.
Brexit minister David Davis has insisted the approach to the European Union divorce had not changed, but at the meeting with lawmakers on Monday, May recognized that a broader consensus needed to be built for Brexit and made clear she would listen to all wings of the party on the issue. "I can't negotiate with myself", Mr Barnier was quoted as saying by the Financial Times.
The talks were "productive", May told a news conference in Paris following a meeting with French President Emmanuel Macron.
Mrs May's largely unchanged Cabinet earlier discussed plans for the scope of negotiations in a meeting that lasted around 90 minutes.
The BBC's political editor Laura Kuenssberg said the prime minister's "colleagues were demanding contrition and sounds like they got it".
Instead, the election stripped May of her majority and obliterated her political authority.
In a surprise move, Michael Gove was appointed environment and agriculture minister less than a year after the prime minister sacked him as justice minister.
A Downing Street source said the talks had been "constructive" but refused to put a timescale on when they would conclude.
On this the government is also facing opposition from the newly strengthened opposition Labour Party.
May called the election saying she needed her own strong mandate to deal with other European Union leaders.
DUP leader Arlene Foster said there had been "positive engagement" so far.
"We achieved what we did last Thursday because we were a united party during the campaign and we need to maintain that unity and collective discipline in the weeks and months ahead", he went on.
"I think there is a unity of goal among people in the United Kingdom".
Democratic Unionist Party (DUP) leader Arlene Foster (L), and DUP Deputy Leader Nigel Dodds prepare to address the media outside Stormont Castle, on the Stormont Estate in Belfast, Northern Ireland, on June 12, 2017.