That will come after the start of the first round of talks about leaving the European Union on Monday.
Adams accused May of playing "fast and loose" with the Good Friday Agreement.
"Any deal that undermines the Good Friday Agreement will be opposed by Sinn Fein and, we would hope, the Irish Government.
"If the institutions are to be put in place they need to be sustainable, viable and properly resourced," he said.
Nigel Dodds, the DUP deputy leader, suggested that any confidence-and-supply deal might not be signed by the time of the Queen's Speech. Sources said it could be signed on Tuesday or Thursday.
This could mean that Brexit talks could begin and Parliament is opened without May being able to agree terms with the DUP to support a minority Conservative government.
With her tally of Conservative MPs reduced to 317 in last week's poll, May needs the backing of the DUP's 10 members to reach the 320 required for a working majority in the House of Commons.
A Tory source said May was "confident" of getting the Queen's Speech through the Commons whether or not a deal was reached with the DUP.
The decision to have it two days later than the scheduled date of June 19 was made after May's regular audience with the Queen on Wednesday.