To stay in government, May must strike a deal with the Democratic Unionist Party (DUP), a small eurosceptic Northern Irish party with 10 parliamentary seats.
Sinn Fein has said the prospect of a deal between the Conservatives and the DUP is causing anxiety and fear, with the party warning it could have implications for power-sharing talks in Northern Ireland.
Sinn Fein has also claimed that a Tory-DUP alliance could damage powersharing talks in Northern Ireland, which has been without an executive since March and without a first and deputy first minister since January.
Also on the cabinet's agenda were the talks to secure an alliance where the DUP backs the Conservatives on a vote-by-vote basis in parliament, rather than a formal coalition government.
It is understood that the Conservatives are seeking a "confidence and supply" deal that would see the 10 Democratic Unionist Party (DUP) MPs lend Mrs May their support for key votes.
Foster said talks with May will continue into next week.
Former British prime minister John Major said the support of the DUP could pitch the province back into turmoil by persuading "hard men" on both sides of the divide to return to violence.
With formal European Union divorce talks due next week, May heads to France on Tuesday to meet Emmanuel Macron, who last month swept to victory in the presidential election.
The Prime Minister has been forced to make assurances to members of her own party on LGBT rights, after prominent interventions from out politicians.
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And he said "appropriate steps" must be taken to ensure that a Labour administration is able to take over negotiations at any stage if Mrs May's Government falls. An agreement to restore devolved power-sharing government in Stormont must be reached by the 29 June deadline.
A senior Conservative source said: "We are making a lot of progress".
But in a statement, officials said: "David Davis, Secretary of State for Exiting the European Union, and Michel Barnier, the European Commission's chief negotiator, agreed today to launch Article 50 negotiations on Monday, June 19".
"If that's not possible, the Labour Party stands ready to offer strong and stable leadership in the national interest".
If that happens, Corbyn will demand a chance to try to form a government by uniting progressive factors in the House of Commons.
Working groups were to take place later to focus on three key areas - the status of European Union citizens living in Britain and British citizens living in the European Union; the divorce bill for Britain; and the status of the Northern Irish border with Ireland.
The BBC reported that a deal with the DUP was expected to be signed on Wednesday.
"Depending on a hard or soft Brexit, that number might be slightly less than that, so it's going to be updated all the time", its United Kingdom chief executive Ian Stuart told BBC television.
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