Defense Secretary Michael Fallon said on the Andrew Marr show he believed the Conservative Party and the DUP will be able to hold together in government.
A stream of Conservative lawmakers walked into 10 Downing St. on Sunday to learn if they had been promoted, demoted or kept in their posts.
May's weakened position in the party rules out big changes to the Cabinet lineup.
As rumours swirled about plots to oust May, Johnson denied he was planning a leadership challenge.
But he has responded in typically colourful fashion, tweeting: "Mail on Sunday tripe - I am backing Theresa may".
She put on a courageous face, refusing to show any contrition for the election gamble that spectacularly backfired, but observers say she has been deeply wounded.
Mr Kenny, the Republic of Ireland's outgoing prime minister, said he wanted to indicate to Ms May his concerns about the impact a Conservative and DUP link-up could have.
May's party had a net loss of 12 seats in Thursday's election, which May called in April in order to bolster her majority status and secure a stronger mandate to begin Brexit negotiations.
When the Conservatives only won 318 of the 326 parliamentary seats they needed for a majority in the House of Commons, this small party enthusiastically emerged as kingmaker.
Members of her Conservative party have warned May her days are numbered after calling Thursday's vote three years early, only to lose her parliamentary majority days before talks on the UK's European Union departure were due to begin.
Many senior Conservatives say May should stay, for now, to provide stability.
Asked whether Brexit negotiations should be delayed, Corbyn told Sky News: "They're going to have to go ahead because Article 50 has been invoked".
As she retreated inside No. 10, it was claimed the Prime Minister had to be told to apologise to her defeated MPs and ministers, by the chairman of the Tory backbench 1922 Committee, Graham Brady. She's then got to present a programme to Parliament.
"As we are the party that won most seats and most votes, we are the only party that is in a position to form a government that can do that".
The Czech prime minister said Friday that Britain should not be granted any extension on the two-year deadline for the Brexit talks.
The heroine of the Conservative campaign, Scottish leader Ruth Davidson, signalled her opposition by tweeting a link to a same-sex marriage lecture she gave a year ago. "As and when details are finalized, both parties will put them forward", a Downing Street spokeswoman said.
The very least which that support could involve would be a confidence and supply arrangement.
The DUP is similar to the "religious right" in the U.S. and takes a hard-line stance on social issues, such as same-sex marriage and abortion.
Protracted negotiations over the formation of a new government - or even a second general election in 2017 - could put back the start of formal talks, squeezing even further the limited time available to forge a complex withdrawal agreement and a separate deal on future trade arrangements.
May spoke to the Irish Taoiseach (Prime Minister) Enda Kenny over the phone on Sunday.
The 1998 Good Friday Agreement commits the United Kingdom government to demonstrate "rigorous impartiality" when dealing with competing political views in the region.
May has insisted that Brexit means Brexit but it is unclear how long she will remain in power or whether another British election will be called.
The debate on the Queen's Speech usually lasts about five or six days.
The Liberal Democrats, whose votes in parliament could help sustain a Labour government, campaigned on the position that Britons should be able to vote again on the terms of the final European Union deal, and stay in the bloc if the deal was rejected.
"We will - obviously - amend the Queen's Speech".
The last count of the election saw Labour overturn a 7,000 majority in Kensington, London, where Emma Dent Coad saw off Tory incumbent Victoria Borwick by just 20 votes after a third recount.
"I think its quite possible there'll be an election later this year or early next year, and that might be a good thing, because we can not go on with a period of great instability", he told the BBC's Andrew Marr on Sunday. "This is still on".