It came as Mr Corbyn vowed to force Theresa May out of Downing Street and to take over as prime minister as a shock new poll showed the party six points ahead of the Conservatives.
Ms May will be forced to rely on support from Northern Ireland's Democratic Unionist Party's (DUP) ten MPs to form a minority Government.
He said: "We haven't won that election".
Instead, voters stripped the Conservatives of their parliamentary majority.
May yesterday lost her two closest aides as she struggled to reassert her leadership after a crushing election setback.
Reports in the Mail on Sunday suggested the foreign secretary was already positioning himself as the successor to May; but he tweeted that the story was "tripe".
Allies said Johnson was spending the weekend resting in Oxfordshire.
Gavin Barwell was named new chief of staff.
The Conservatives won 318 House of Commons seats in the election, eight short of an outright majority.
The change was unlikely to significantly quell unrest within the party. The new membership applications have brought total membership numbers to 800,000.
"This is still on", he said, adding he would vote down the government's programme when it comes before parliament this month.
The Labour leader insisted he can still form a government - despite losing last week's vote getting only 262 seats.
"The DUP has been part of the parliamentary process, and the democratic process in Northern Ireland for many years". It aligns London more closely with the pro-British side in the divided province, where a power-sharing government with Irish nationalists is suspended.
The turmoil engulfing May has increased the chance that Britain will fall out of the European Union in 2019 without a deal. Their reportedly caustic style made them enemies and the disastrous election result made them vulnerable.
May also undertook to keep senior ministers, saying they would remain in their posts, despite rumours that she had been ready to sack Finance Minister Philip Hammond following a clash over her Brexit strategy. She confirmed this to German leader Angela Merkel in a phone call on Saturday.
Foster is due to travel to London on Tuesday to meet May to discuss the details of a possible arrangement.
He said he believed there was a majority in the Commons for such an approach.
Mr Corbyn, who has faced opposition from within his party since becoming leader in 2015, said he believed Labour was now more united.
"She's staying, for now", one Conservative Party source told Reuters.
The Times newspaper's front page declared that Britain was "effectively leaderless" and the country "all but ungovernable". Later Conservative MP Dominic Raab said the deal would be published.
As the former Tory minister Nicky Morgan told me this morning, the proposal to allow many more grammar schools to be created looks dead buried.
Speaking on Marr, Sir Michael said the government wanted maximum access to the European Union single market and an "arrangement on immigration".
Spokesperson for the Office of the British Prime Minister (10 Downing Street ) said that Brexit talks will go ahead as planned.