A senior Conservative source told AFP that the party was "confident" it would have enough parliamentary votes to pass the Queen's Speech, but the deal with the DUP has not been finalised yet.
Commons leader Andrea Leadsom said the government had "agreed with Buckingham Palace" to go ahead with the State Opening of Parliament, a constitutional requirement before MPs can hold debates and begin work.
The Queen's Speech was originally meant to take place on Monday 19 June, but was delayed because of the governments's discussions with the DUP in order to secure a "supply and confidence" agreement.
The State Opening of Parliament by the Queen is to take place on June 21, after being delayed by two days by the inconclusive outcome of last week's General Election.
While talks on a long-term deal to keep the Conservatives in power are ongoing, the government wants to get on with its business, the official said.
Others, such as former premier Sir John Major, have warned that any formal arrangement would undermine the fragile Northern Ireland peace process.
With eight seats short of an overall majority, the co-operation from the 10 DUP members will ensure May has a majority to see through her program of measures in the next session of parliament.
"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.
The meetings came as time is running out for the parties to reach an agreement on restoring devolved government to Stormont, with the Government set to resume direct rule from Westminster if there is no deal by June 29.
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.
"We have two weeks from today to get the Northern Ireland executive up and functioning again, and to try to bring in a new chair is actually a waste of time and a distraction", said Mr Swann.