However, the prospect of a deal between the two parties has caused disquiet, with the DUP's anti-abortion and gay rights stance in the crosshair.
May's party fell eight seats short of retaining its parliamentary majority, and is now in talks with Northern Ireland's ultra-conservative Democratic Unionist Party (DUP) - which won 10 seats-to forge an informal alliance.
A government official said the talks were proceeding well and Foster tweeted that "we hope soon to be able to bring this work to a successful conclusion".
Prime Minister Theresa May has made a joke at her own expense as Britain's House of Commons got underway - a reflection of her new humility following a disastrous snap election in which she lost her majority.
"There can be changes in the offer of Brexit as we go forward", Davidson told reporters in London after meeting May.
Labour's unexpectedly strong second-place showing has thrown national politics into disarray. "The union as I've said before is our guiding star", she said.
"Obviously until we have that we can't agree the final details of the Queen's Speech", said May's deputy Damian Green, referring to an agreement with the DUP. During the election campaign, May had used the "coalition of chaos" phrase to describe what a victory for Labour would look like.
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The process resumed on Monday after a pause for the General Election, but negotiations have again been put on ice on Tuesday as the political focus shifted to Westminster and the anticipated parliamentary deal between the DUP and Conservatives.
After attending the Prime Minister's political cabinet on Monday, she told BBC News: "I'm suggesting that the Conservative Party works with those both within the House of Commons and with people without to ensure that as we leave the European Union we have a Brexit that works for the economy and puts that first".
She also signalled she's willing to rethink her approach to Brexit.
The Evening Standard edited by Tory former chancellor George Osborne, and the Daily Telegraph, have reported that Cabinet ministers have initiated talks with Labour MPs to secure cross-party backing for a softer Brexit.
In a surprise move, Michael Gove was appointed environment and agriculture minister less than a year after the prime minister sacked him as justice minister.
Eurosceptic Conservative MP John Redwood said: We're going to make a great offer to Europe in terms of their access to us and continued presence of their citizens here in Britain - they are very welcome.
Foster will nearly certainly ask for greater investment in Northern Ireland as part of the deal, as well as guarantees on support for pension plans and for winter fuel allowances for older people.
Don't waste time, European Union Brexit chief Michel Barnier tells Britain
Pulling together a government is proving to be quite the headache for Theresa May . Her performance was roundly praised by Conservative MPs.
A majority of people in Northern Ireland also voted to remain in the European Union in last year's Brexit referendum.
They are adamant the British government can no longer cast itself as a neutral facilitator in the process, given the prime minister's likely deal with the DUP.
A deal with the DUP could also risk destabilizing the political balance in Northern Ireland by increasing the influence of pro-British unionists who have struggled for years with Irish Catholic nationalists who want Northern Ireland to join a united Ireland.
'Any deal which undercuts the Good Friday and other agreements has to be opposed by progressives'.
The stakes for May are high. Labour Leader Jeremy Corbyn has close ties to Sinn Fein and the Sinn Fein MPs are understood to be hoping that in return for their support, Mr. Corbyn would back a call for a referendum on Northern Ireland leaving the United Kingdom and joining Ireland. If that happens, Corbyn will demand a chance to try to form a government by uniting progressive factors in the House of Commons.
Scottish First Minister Nicola Sturgeon said May's government lacked the credibility necessary for Brexit talks and should delay the negotiations.
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