Theresa May has signalled she is "confident" of getting the Queen's Speech through the Commons whether or not a deal is reached with the Democratic Unionists (DUP) by the State Opening of Parliament next week.
Ms Foster said the talks deadline remained "realistic".
Amid calls from some MPs for the Conservatives to rethink their Brexit strategy, he said there was a "clear consensus" for leaving the single market and ending free movement while retaining the "maximum access" to European Union markets and maintaining co-operation in key areas such as science.
The leader of the Scottish Conservatives Ruth Davidson, who is a lesbian, had spoken out on the issue after telling the Prime Minister that she considers equality to be more important than party politics.
William Hague, a former leader of the Conservative Party, called for business groups and lawmakers from all parties to be brought in to agree a national position on Brexit.
Sinn Féin President Gerry Adams and the party's leader in Northern Ireland Michelle O'Neill will be in London on Thursday to express concerns that a Tory agreement with the DUP will upend attempts to restore the power-sharing agreement, the Independent reported.
"There is a unity of objective among people in the United Kingdom", May said following a meeting with French President Emmanuel Macron in Paris.
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.
During the campaign, May cast herself as the only leader competent enough to navigate the tortuous Brexit negotiations that will shape the future of the United Kingdom and its $US2.5 trillion ($A3.3 trillion) economy.
Tories have made clear since last week's election that their discussions with the DUP revolve around assurances of support in key Commons votes, rather than a full coalition.
"They can't have it both ways, it has to be dealt with sensibly", she said.
He told BBC Radio 4's World At One Programme: 'People regard the peace process which was very hard earned over very many years by a lot of people, people shouldn't regard it as a given, it isn't certain, it is under stress, it is fragile. I am wary about it. "I am dubious about it", Major said.
Brexit Secretary David Davis and the European Commission's chief negotiator Michel Barnier agreed during discussions in Brussels on Thursday to start formal talks over the UK's departure on 19 June.
While the DUP are deeply eurosceptic, they have baulked at the potential loss of a "frictionless border" with the Republic of Ireland.
"We need to do everything we conceivably can to ensure that doesn't happen - and that does require an impartial British government", Sir John said.