Foster added: "I know people want to talk about soft Brexit, hard Brexit, all of these sorts of things, but what we want to see is a sensible Brexit and one that works for everybody".
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.
Theresa May has signalled she is "confident" of getting the Queen's Speech through the Commons whether or not a deal is reached with the Democratic Unionists (DUP) by the State Opening of Parliament next week.
"She said: "I'm the person who got us into this mess and I'm the one who is going to get us out of it", said one Conservative lawmaker who attended Monday's meeting. Critics of the general election point to the Conservative's policies of austerity as a key reason they lost so many seats in parliament, so sending extra money to Northern Ireland when it's desperately needed at home will not go down well with voters.
Earlier this week, in a joint press conference with Mrs May, French president Emmanuel Macron claimed the door is always open for the United Kingdom to remain in the European Union before Article 50 negotiations are concluded.
After London fire, desperate pleas for missing residents
Mr Alhajali's older brother Omar - who was with him in the flat - survived the fire after they were separated on the way out. She had no phone to call for help in the early hours of Tuesday morning, which is why she had taken to Facebook, said Lammy.
Her failure to win a majority has put May under pressure over her Brexit plans from inside and outside her party and has prompted complaints about her choice of partner due to the DUP's stance on social issues such as gay marriage.
These concerns were echoed by former prime minister Sir John Major who has warned that the Government will compromise its stated impartiality if it enters a confidence and supply deal with the DUP at Westminster.
"The door of course is still open as long as Brexit negotiations have not been concluded", Mr Macron said.
Despite the uncertainty over her ability to govern, May had confirmed that Brexit negotiations - expected to be the most complex global talks Britain has held for decades - would begin as planned next week.
Nevertheless, it illustrated the challenge Prime Minister May will face - finding a position that satisfies both pro-European and eurosceptic factions of her party - if she wants to remain in power.
Under the proposed deal, the DUP would likely support May's Conservatives on big issues such as the budget, Brexit and defence legislation on a vote-by-vote-basis.