A "Brexit election" that left UK Prime Minister Theresa May without an overall majority has forced her into a precarious alliance with the far-right Democratic Unionist Party, who are deeply opposed to gay rights and abortion.
It is thought Mrs Foster, despite being a Brexit supporter, could seek assurances from Mrs May that she will pursue a softer exit from the European Union, given Northern Ireland's 56% Remain vote and the DUP's desire not to see a return to a hard border with Ireland.
May is set to meet DUP leader Arlene Foster in London on Tuesday but will face demands from the DUP for more money for Northern Ireland.
Leave campaigner Michael Gove has indicated the Government is ready to pursue a softer Brexit deal and work with Labour to get it through Parliament.
But the Government will press ahead with the first round of Brexit talks two days before the official opening of Parliament. The finance ministry declined to comment.
In a hint at a softer approach, Mr Davis said on Monday "we will start down this process" by focusing on the divorce proceedings from the European Union before moving on to trade.
The Conservatives, who have 317 MPs, are looking to strike a deal with the DUP, who have 10.
While a deal with the DUP would help May stay in power to open Brexit talks that are due to begin next week, it would also risk destabilizing the political balance in Northern Ireland by increasing the influence of pro-British unionists.
He rightly pointed out that if the Northern Irish people no longer see the United Kingdom government as an honest broker in the ongoing unionist-nationalist debate, "one can't be certain how events will unwind".
"I am dubious about it", Major, who served as prime minister from 1990 to 1997, told BBC radio.
The Prime Minister met with representatives of all five main political parties from the province with a view to getting the Assembly back up and running again.
May faces a hard balancing act.
Divisions over Europe helped sink the premierships of Ms May's predecessors Margaret Thatcher, John Major and Cameron, and many of her lawmakers and party members favour a sharp break with the EU. Government bond prices suffered heavy losses on Tuesday after consumer price inflation jumped to 2.9 percent in May.
Like many on the continent, he has long criticised Britain's entitlement, negotiated by Prime Minister Margaret Thatcher in the 1980s, to get about half its European Union budget contributions back as a rebate.