A week after May lost her majority in an election she had called in the hope of strengthening her hand in the talks, some fellow Conservatives want her to focus more on limiting the damage to business and less on cutting immigration and other ties to the European Union when Britain leaves the bloc in March 2019.
"We continue to work with all the parties in Northern Ireland and the Republic of Ireland in ensuring that we can continue to put in place those measures necessary to fulfil those agreements, " she said.
"Look at what the Tories (Conservatives) have managed to do to the United Kingdom in the space of just one year, firstly calling a divisive and reckless European Union referendum. then having lost that gamble pursuing a hard Brexit path", Scotland's first minister, Nicola Sturgeon, said.
Jeremy Corbyn, leader of the opposition Labour Party, which saw its number seats and share of the vote increase, said there could be another election this year or early in 2018 after last Thursday's vote produced no clear victor.
"I congratulate her on returning and I'm sure she'll agree with me that democracy is a wondrous thing, and can throw up some very unexpected results", he said.
The Irish nationalist party Sinn Fein, which wants the British province to join a united Ireland, said the prospect was causing anxiety and fear in Northern Ireland.
The veteran conservative predicted that Britain would regret its departure from the bloc at some point in the future. Whatever her ultimate plan, she will be heavily reliant upon the 10 lawmakers from the eurosceptic DUP, who would help her edge past the 326 votes needed in parliament to avoid the government collapsing.
"Discussions are going well with the government", Foster said.
The Conservative source said talks with the small Northern Irish party were progressing well ahead of meetings later Thursday in Downing Street with all of the British province's main political leaders.
For example, the DUP blocked same-sex marriage being implemented in Northern Ireland and has stood in the way of abortion rights being extended to women there.
He also suggested that the Scottish Tories led by Ruth Davidson could also add to the pressure on Mrs May to change course.
Foster's rivals in Northern Ireland, such as Sinn Fein's Gerry Adams, have objected.
The Commission has stated that one of its priorities in Brexit talks was the Good Friday Agreement, which brought an end to violence and terrorism in Northern Ireland in the late 1990s.
Some opposition politicians say that stance is no longer acceptable, with some in her own party pressing May to soften her position and focus on saving jobs after Britain leaves.
The stakes are high: 44 percent of British exports in 2015 went to the European Union while 53 percent of imports came from the bloc.
Among these was whether Britain's position would "be the same as in the letter of March 29" when May triggered the two-year Brexit countdown.
"It's passing quicker than anyone believes..."
"Whether they're devolved or not as issues, the problem people over here have is what kind of mentality do they have, that Theresa May is prepared to rely exclusively on their support now to run a government over here?"
A Conservative source said there was so far no deal to announce and that a decision on the timing of any announcement would have to wait until an agreement was finalised. "I can't negotiate with myself", he told The Financial Times.