Talks between the Tories and Democratic Unionist Party aimed at shoring up Theresa May's position in Parliament were continuing on Wednesday.
The Prime Minister needs the votes of the 10 DUP MPs to prop up her minority administration as she hopes to steer government business - including crucial measures on Brexit - through the Commons.
Mrs May is coming under intense pressure to change her approach to leaving the European Union, with predecessor David Cameron among those suggesting a softer stance with a greater effort to seek a consensus.
Talks with the DUP to secure her government broke up on Tuesday night without an agreement, but Mrs May said the discussions had been "productive".
A senior Conservative source said: "We are making a lot of progress, it's all being done in the spirit of cooperation, with a real focus on strengthening the union and providing stability at this time."
DUP leader Arlene Foster, who travelled to Westminster for talks with the Tories on Tuesday, said she hoped a deal could be reached "sooner rather than later".
The "supply and confidence" deal would see the DUP back the Conservatives in votes on the Budget and confidence motions.
One of the most pressing issues facing Mrs May is the process of leaving the European Union, with Brexit talks set to start in Brussels on Monday.
Former prime minister Mr Cameron said Mrs May would have to change her approach to Brexit as a result of the election.
"It's going to be difficult, there's no doubt about that, but perhaps an opportunity to consult more widely with the other parties on how best we can achieve it," he said at a conference in Poland, the Financial Times reported.
"I think there will be pressure for a softer Brexit," Mr Cameron added, saying that Parliament now "deserves a say" on the issue.
He also suggested that the Scottish Tories led by Ruth Davidson could also add to the pressure on Mrs May to change course.
"There's no doubt that there is a new player on the stage," Mr Cameron said. "Scotland voted against Brexit. I think most of the Scottish Conservatives will want to see perhaps some changes with the policy going forward."
The Times reported that Chancellor Philip Hammond was preparing to fight for the UK to remain within the EU's customs union.
That would help safeguard jobs and trade with EU members, but would severely restrict the UK's ability to strike its own trade deals around the world.
At a joint press conference with Mrs May in Paris on Tuesday night, French President Emmanuel Macron suggested that the door was still open for the UK to remain within the European Union.
Mr Macron said: "Of course the door remains open, always open until the Brexit negotiations come to an end."
He added: "That being said a sovereign decision was taken by the British people and that is to come out of the European Union and I very much respect the decisions taken by the people, be it by the French people or the British people.
"As a matter of fact in this case it's not for me to say whether or not this decision should be questioned - the decision to leave the European Union - but until the negotiations come to an end, of course there is always the possibility to re-open the door.
"But let us be clear and organised and once the negotiations have started we should be well aware that it'll be more difficult to move backwards."
Mrs May stressed that Brexit would happen and the timetable remains on course.
She said: "I think there is a unity of purpose among people in the United Kingdom.
"It's a unity of purpose, having voted to leave the EU, that their Government gets on with that and makes a success of it, and we are committed to developing a deep and special partnership with the EU.
"We want the EU to continue to remain strong and we want to continue to cooperate."
Shadow business secretary Rebecca Long-Bailey raised concerns over the prospect of a "very dangerous deal" between the DUP and the Tory Party, saying Labour was "ready and waiting to form a minority government".
She dismissed accusations of hypocrisy after it emerged that Gordon Brown had tried to do a deal with the DUP in 2010, telling BBC Radio 4's Today programme: "It's certainly not something that Jeremy would advocate and the Labour Party is certainly not advocating that.
"As I've said, it would create a lot of instability in terms of the peace process in Northern Ireland and it's a very worrying time.
"It just illustrates the chaos that the Conservative Party are in at the moment."
Ms Long-Bailey refused to rule out retaining freedom of movement in exchange for staying in the single market after Brexit, as she urged the Prime Minister to seek consensus with other parties.
She said: "We accept the fact that, if we are going to have impediment-free access to the single market, then there will have to be some element of free movement.
"I think we know that, the Conservative Party know that, but the specific points are open to negotiation."