The EU "would be willing to renegotiate" a Brexit deal, says Tory leadership hopeful Jeremy Hunt, adding "they want to solve the problem".
The EU has previously said the withdrawal agreement reached with the UK cannot be reopened.
Unlike the race frontrunner, Boris Johnson, Mr Hunt did not commit to leaving the EU on 31 October.
Meanwhile, fellow leadership contender Rory Stewart insisted "there is no new negotiation with Europe".
He said the EU had made it clear they would not revisit the withdrawal agreement. Instead he proposed setting up a citizens' jury to break the Brexit impasse.
Under his plan, a group of 50,000 people would be selected randomly from the electoral register. Those people would get a phone call in late July to check they were available to participate.
A polling company would then whittle the number down, making sure the final group was representative of the country. That group would be given three weeks to make recommendations which Parliament would then be able to approve or reject.
Dominic Raab - also running to replace Theresa May and become the next prime minister - told Sky News' Ridge on Sunday programme the Conservative Party "will be toast unless we are out by October".
The UK had been due to leave the EU on 29 March, but EU leaders agreed to delay the date, after MPs repeatedly rejected Theresa May's Brexit deal.
The current date for leaving the EU is 31 October.
Where do the candidates stand on Brexit?
- Michael Gove has said he would negotiate changes to the Irish backstop and would accept a "short delay" to Brexit to achieve a deal
- Jeremy Hunt also wants to negotiate a new deal with changes to the backstop. He says he would accept a no-deal Brexit if "there is no prospect of a deal by 31 October"
- Sajid Javid has proposed "a new digitised" Irish border which would not involve any infrastructure on the border. He stated he cannot envisage wanting to extend the UK's exit date
- Boris Johnson has said the "way to get a good deal is to prepare for no deal" and has committed to leaving on 31 October "deal or no deal"
- Dominic Raab wants to re-open the withdrawal agreement but has also argued that leaving without a deal "is far better than leaving with a fatally flawed deal"
- Rory Stewart prefers trying to push through the current deal, agreed by Theresa May. However he says, if that failed, he would turn to the option of a "Brexit assembly" of citizens to thrash out a compromise
Mr Hunt told the BBC's Andrew Marr Show the EU is open to solutions surrounding the Northern Ireland backstop.
"They say if they were approached by a British prime minister who had ideas on how to solve the Northern Ireland backstop, they would be willing to renegotiate the package."
He also said it would be wrong to commit to leaving the EU by 31 October, but added "if there was no prospect of a deal," he would be prepared to leave the EU without a deal.
Keeping up with Johnson
Analysis by BBC political correspondent Nick Eardley
With Boris Johnson so far ahead when it comes to support from Tory MPs, the other candidates are increasingly pitching to be the other person on the ballot of Tory members.
Jeremy Hunt's claim he can renegotiate the deal will seem overly optimistic to many, and completely impossible to some.
His refusal to guarantee the UK will leave this year will also concern many Tories - who worry about the process going on and on and on.
Hence Dominic Raab's warning his party will be toast unless it delivers in October.
But listen carefully and he's also turning his fire on Mr Johnson - questioning whether the frontrunner has a proper plan on Brexit.
That point is made much more directly by Rory Stewart who says simply that he doesn't think Mr Johnson can deliver.
In this race, it's fast becoming about how to stay in the race with Mr Johnson - even if that involves trying to trip him up.
Mr Hunt said Boris Johnson was "effectively committing the country to no-deal or an election" by saying he would definitely leave the EU on 31 October,
The foreign secretary also said he had "profound issues" with Theresa May's approach to getting a deal through Parliament.
"I did not think we should be trying to persuade Parliament to accept the backstop," he said.
'It begins to come off the rails'
Dismissing the idea of getting new deal from the EU, Mr Stewart said other candidates "who are promising what they can't deliver are going to let people down terribly".
He said the tactic of threatening no deal in order to secure a better deal with the EU was unrealistic.
"The EU is not scared of it because it is not a credible threat," he said.
He challenged Mr Johnson - who won support from 114 MPs in the first leadership ballot - to reveal his Brexit tactics in the BBC debate on Tuesday.
"As soon as I sit down with him and ask how are you going to deliver Brexit, then it begins to come off the rails," he told Marr.
Mr Stewart has ruled out serving in Boris Johnson's cabinet, if the ex-London Mayor becomes prime minister.
On Tuesday 18 June BBC One will host a live election debate between the Conservative MPs who remain in the race.
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