Earlier in the day ministers confirmed the date of the Queen's Speech, with Leader of the Commons Andrea Leadsom saying: "The Government has agreed with Buckingham Palace that the State Opening of Parliament will take place on 21 June 2017".
Theresa May announced her plan to lead a minority government after the election result with the support of the DUP - though the terms of any arrangement with the Northern Irish party have yet to emerge.
A senior Conservative source briefed journalists that there was "broad agreement on the principles of the Queen's Speech", which will set out a slimmed-down programme for government following the failure to secure a majority last week.
Timing suggests Tories and DUP have reached a deal.
It is not yet clear whether the European Union withdrawal talks will go ahead on that day, although Brexit Secretary David Davis has said they will start "next week".
The Prime Minister was meeting separately with representatives of Sinn Fein, the Ulster Unionists, the SDLP and the Alliance Party, as well as the DUP, in Downing Street in an attempt to allay growing concerns.
Foster had previously said an alliance between the DUP and Conservative Party would help defend Northern Ireland's position as the United Kingdom prepares to begin Brexit negotiations, which are due to kick off next week.
The 1998 Good Friday Agreement, also referred to as the Belfast Agreement, requires the United Kingdom and Irish Governments to show "rigorous impartiality" as they deal with the different political groups in Northern Ireland.
"We are working on a series of fronts and we are working very, very hard to deliver good government for Northern Ireland and good government for the United Kingdom", he said.
DUP deputy leader Nigel Dodds, in Downing Street for talks with Mrs May on restoring powersharing in Northern Ireland, refused to be drawn on how the negotiations were progressing.