Clarke also backed May remaining in Number 10 despite her failure to retain the Conservative Party's parliamentary majority at Thursday's general election.
The Daily Telegraph reported cabinet ministers have opened back-channel talks to senior Labour lawmakers to secure a cross-party agreement on Brexit.
Ms May made clear the Brexit negotiations would begin on schedule despite the political uncertainty at home.
One of British Prime Minister Theresa May's predecessors has warned that her plan to stay in power after losing her parliamentary majority risks the "fragile" peace in Northern Ireland.
Despite her party's expectations of a landslide victory, May lost her majority in parliament, pushing her into rushed talks on a support agreement with a small Eurosceptic Northern Irish Protestant party with 10 parliamentary seats.
"I congratulate her on returning and I'm sure she'll agree with me that democracy is a wondrous thing, and can throw up some very unexpected results", he said.
Brexit minister David Davis has insisted the approach to the European Union divorce had not changed, but at the meeting with lawmakers on Monday, May recognised that a broader consensus needed to be built for Brexit and made clear she would listen to all wings of the party on the issue.
Michael Gove, former Brexit leave campaigner and now Environment Secretary, later confirmed to BBC television that he had discussed Brexit with Labour politicians.
A contrite May took responsibility for losing her party's majority; 'I've got us into the mess and I will get us out of it.' She also promised to help those who had lost their seats and said sorry a couple of times.
Clarke backed Remain in last year's European Union referendum, but said that "the parliamentary vote in the last parliament settled the doubt we are leaving. we are now destined to leave the European Union".
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The Irish republican Sinn Fein party - which won seven seats in the election, although their MPs traditionally do not take up their seats in protest - is also wary of the alliance.
A poll released last summer has shown 70% of people in Northern Ireland support same-sex marriage.
Even the idea of an alliance is complicated, however.
The unionists have struggled for years with Irish Catholic nationalists who want Northern Ireland to join a united Ireland.
Foster's rivals in Northern Ireland, such as Sinn Fein's Gerry Adams, have objected, describing any partnership between the Conservatives and the DUP as "a coalition of chaos".
But former Conservative Prime Minister John Major said he was "dubious" about a deal and its impact on the peace process.
"My preoccupation is that time is passing, it is passing quicker than anyone believes because the subjects we have to deal with are extraordinarily complex". "We are united in our total condemnation of terrorism and our commitment to stamp out this evil", May said ahead of her visit. "I can't negotiate with myself".
"There is wide spread concern that Theresa May in seeking a deal with the DUP to remain in office will make the job of re-establishing the Executive more hard".
The EU's negotiator Michel Barnier dismissed the suggestion of postponing the negotiations and said such a delay would only prompt further instability.
Mr Barnier lamented that it was already three months since Mrs May had formally triggered the two-year process of Britain leaving the European Union.
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