But Mr Corbyn said: "I don't think Theresa May and this government have any credibility".
May earlier on Saturday lost her two closest aides as she struggled to reassert her leadership after a crushing election setback.
May, left, and Arlene Foster, leader of the Democratic Unionist Party, have effectively formed an alliance.
"Our view of Brexit I don't think has changed".
In a WhatsApp message to Conservative lawmakers, Johnson said: "Folks we need to calm down and get behind the prime minister". Gavin Barwell was named new chief of staff.
May's weakened position in the party ruled out big changes.
Overnight, the government was thrown into confusion after the DUP contradicted a Downing Street statement on Saturday that said a "confidence and supply" agreement had been reached with the DUP and would be put to the cabinet on Monday. It aligns London more closely with the pro-British side in the divided province, where a power-sharing government with Irish nationalists is suspended.
On Brexit, Mr Corbyn said he wants a "jobs-first Brexit" negotiated as quickly as possible along with guaranteeing the post-Brexit rights of European Union nationals living in the UK. "Absolutely", Corbyn told Sunday Mirror newspaper in an interview.
The British pound tumbled on Friday against the US dollar and the euro before stabilising, down 1.7 and 1.4 percent against the two currencies respectively.
May has said Brexit talks will begin on June 19 as scheduled, the same day as the formal reopening of parliament.
"We welcome this commitment, which can provide the stability and certainty the whole country requires as we embark on Brexit and beyond", he said.
'Only Boris Johnson or David Davis can be next Tory leader'
One ally told the Sunday Times: "We are facing a populist and they have realised we need someone who can talk to the people. We need a Brexiteer", they told the Mail on Sunday .
"She's then got to present a programme to Parliament".
Britain's typically pro-Conservative press questioned whether Mrs May could remain in power with the clock ticking on the two-year European Union divorce process.
"Moderate" MPs last night admitted it was time for a unified front from a parliamentary party that has been split in two ever since Jeremy Corbyn took over in 2015.
It comes after Mr Corbyn confirmed he would offer a substantial amendment to the Queen's Speech that he will attempt to push through Parliament.
"She's staying, for now", one Conservative Party source told Reuters.
Some say her failure means the government must now take a more flexible approach to the divorce, potentially softening the exit terms.
The former shadow chancellor labelled it an "OK result" after Mr Corbyn's party secured a higher-than-expected 262 seats and significantly boosted its vote share. "May stares into the abyss", wrote The Times, while the Conservative-supporting Sun tabloid said succinctly: "She's had her chips".
"The Conservatives have not yet broken the British system of democracy, but through their hubris and incompetence they have managed to make a mockery of it", it said in an editorial. Instead, Cameron still would be prime minister, presiding over a majority government (however narrow) of a United Kingdom that was still part of Europe. On the other hand, pro-Europe Scottish Conservative leader Ruth Davidson said she wanted to be involved in "looking again" at Britain's aims for Brexit.
Membership soared following Friday's result, according to Labour MP Richard Burgon, which saw Theresa May's Conservatives fail to win an outright Commons majority.
"Imagine she survives until autumn of next year", he said.
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