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A no-deal Brexit would be a failure that both the British and Irish governments would be responsible for, Boris Johnson has said.
The prime minister was in Dublin for his first meeting with Irish PM Leo Varadkar since he entered Number 10.
The government has confirmed Parliament will be suspended later after a vote on holding an early general election.
Opposition parties will not back the vote, meaning there will be no election in October as the PM had hoped.
Mr Johnson said he believed a Brexit deal was still possible by the EU summit in October, but Taoiseach Varadkar said there was no such thing as a "clean break" between the UK and the EU.
Mr Johnson has ruled out asking the EU to delay the Brexit deadline of 31 October, although the Irish government has said it would support another extension.
Ahead of their meeting, Mr Varadkar said he would be asking his UK counterpart how he plans to get a Brexit deal through Parliament without a majority in the House of Commons
But Mr Johnson told reporters in Dublin he was "absolutely undaunted" about what might happen in Parliament.
Mr Johnson also said he had looked at the consequences of a no-deal Brexit, and "wanted to find a deal".
"We must restore Stormont and we must come out [of the EU] on 31 October, or else permanent damage will be done in the UK to trust in our democratic system," he added.
The two leaders spoke privately for half an hour before joining their delegations for another half-hour meeting, said a joint statement.
"While they agreed that the discussions are at an early stage, common ground was established in some areas although significant gaps remain," it added.
By Jayne McCormack, BBC News NI Political Reporter
Boris Johnson arrived in Dublin trying to strike a more conciliatory tone than in previous weeks.
Perhaps brought to bear by cabinet ministers concerned about no deal, Mr Johnson insisted he wants an agreement.
But trust is a crucial tenet of any political relationship.
A stony-faced Leo Varadkar said a deal is still possible, but warned that promises from the UK wouldn't cut it.
The Irish government is also watching Westminster events closely, knowing the prime minister is currently boxed in by opposition parties, with few options left.
A breakthrough was never going to happen today, but what goes on behind closed doors around this key meeting could potentially pave the way for movement.
The taoiseach invited Mr Johnson to Dublin two months ago, shortly after he was installed in Downing Street, to discuss Brexit.
The Irish government maintains that the backstop - the mechanism to avoid an Irish hard border - is needed in any withdrawal agreement, because of decisions made by the UK.
But Mr Johnson has said he will not sign up to a deal unless the backstop is removed, because it is "anti-democratic".
Ahead of the visit to Dublin, the prime minister said he wanted to discuss the possibility of an all-Ireland food standards zone as part of a solution to replace the Brexit backstop.
There is also speculation that the government could propose returning to a backstop that would only apply to Northern Ireland, with the possibility of a role for the Stormont assembly before it could be triggered or new EU rules would take effect.
Meanwhile, Northern Ireland Secretary Julian Smith has denied claims he would leave his position, following the resignation of Amber Rudd as pensions secretary at the weekend.
Parliament will be suspended when business in the Commons finishes later on Monday, after MPs have voted again on whether to hold an early general election.
Before Parliament is prorogued, MPs will debate progress reports updating them on efforts to restore the Stormont assembly.
Northern Ireland's devolved parliament collapsed in January 2017 when Sinn Féin's Martin McGuinness resigned as deputy first minister, citing the DUP's conduct around a flawed green energy scheme..
Unless power-sharing talks succeed before Brexit happens, Mr Smith has said direct rule powers from Westminster will need to be implemented at pace.
Meanwhile, legislation designed to delay a no-deal Brexit and force the prime minister to request an extension to the EU deadline will receive royal assent later.
However, Mr Johnson has said he will not ask the EU for another extension, so it is unclear what might happen next.
The backstop is a key piece of the Brexit deal dictating what will happen to the border between Northern Ireland and the Republic of Ireland.
It is a last resort that guarantees a frictionless border if no better solution is devised in time - by maintaining close ties between the UK and the EU until such a solution is found.
The Irish government has insisted if it took effect it would only be temporary, but Mr Johnson has said he will not sign up to a deal unless the backstop is removed from the withdrawal agreement.
The government will move another motion asking MPs to vote for a general election on Monday too, but it is unlikely to pass because opposition parties have agreed to reject the demand, saying Mr Johnson is trying to force through a no-deal exit.
