The prime minister is in Dublin for his first face-to-face meeting with Taoiseach (Irish PM) Leo Varadkar since he entered Downing Street in July.
He said he believed a deal was still possible by the EU summit in October.
However Mr 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 – but the Irish government said it would support another extension.
Earlier on Monday Irish Finance Minister Paschal Donohoe said an extension would “create space”.
But he said it was a matter that needed “full agreement” across the European Council. Under EU rules all member countries – including the UK – must agree to an extension.
Mr Varadkar has also warned the risk of no deal is “significant and growing”.
Speaking during a press conference at Dublin Port on Sunday, Mr Varadkar said he would be asking Boris Johnson how he plans to get a Brexit deal through Parliament when he does not have a majority in the House of Commons.
Why is the meeting taking place?
The taoiseach invited Mr Johnson to Dublin two months ago, shortly after he was installed in Downing Street, to discuss Brexit.
The two leaders have very different views on how the deadlock should be resolved.
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 would use the meeting to raise potential alternatives with Mr Varadkar.
But Dublin has said the UK has not yet offered any “credible” alternative to the backstop and insists it only negotiates through the European Union’s team.
Meanwhile, Northern Ireland Secretary Julian Smith has denied speculation that he would leave his position.
In a tweet, on Monday morning, he said he intended to continue to “work flat out” for Northern Ireland.
Will Parliament be suspended this week?
It is due to happen at some stage between Monday and Thursday and MPs are not set to return until mid-October.
Before Parliament is prorogued, MPs will debate progress reports updating them on efforts to restore the Stormont assembly, which collapsed more than two years ago.
The government has said it is planning to “intensify” efforts to kick start power-sharing, because there are concerns about the impact of prorogation on Northern Ireland.
The Northern Ireland Secretary Julian Smith has said unless power-sharing talks succeed before Brexit happens, 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 deadline from the EU will receive royal assent later.
However, the prime minister has said he will not ask the EU for another extension, so it is unclear what might happen next.
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.
What is the backstop?
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 Boris Johnson has said he will not sign up to a deal unless the backstop is removed from the withdrawal agreement.
He said he wanted to discuss the possibility of an all-Ireland food standards zone as part of a solution to replace the Brexit backstop.