Boris Johnson has told the EU the Irish border backstop must be scrapped as it is "unviable" and "anti-democratic".
In aÂ letter to European Council President Donald Tusk, the PM also said the backstop risked undermining the Northern Irish peace process.
But Mr Tusk said those opposing the arrangement without "realistic alternatives" supported re-establishing a hard border on the island of Ireland.
This was the reality "even if they do not admit it", he added.
"The backstop is an insurance to avoid a hard border on the island of Ireland unless and until an alternative is found," Mr Tusk tweeted.
Two possible outcomes?
The EU has long insisted the withdrawal agreement, including the backstop, cannot be reopened. It was negotiated by former PM Theresa May but has been rejected three times by MPs.
In a phone conversation with Mr Johnson on Monday evening, Irish Taoiseach (prime minister) Leo Varadkar reiterated that the agreement was closed and the backstop must stay.
Mr Johnson's four-page letter to Mr Tusk came ahead of meetings this week with German Chancellor Angela Merkel and French President Emmanuel Macron.
The BBC's political correspondent Iain Watson said Mr Johnson appeared to be aiming for one of two outcomes - either the EU blinks as the prospect of no deal looms ever closer, or it holds firm, in which case the PM will argue it is Brussels' fault the country is heading for no deal.
The border is a matter of great political, security and diplomatic sensitivity, and both the UK and EU agree that whatever happens after Brexit there should be no new physical checks or infrastructure at the frontier.
The backstop is a position of last resort to guarantee that, but if implemented, it would see Northern Ireland stay aligned to some rules of the EU single market.
It would also involve a temporary single customs territory, effectively keeping the whole of the UK in the EU customs union.
In his letter, Mr Johnson described the backstop as "inconsistent with the sovereignty of the UK".
He said the arrangement would be "anti-democratic" because it offered no means for the UK to unilaterally exit and no say for the people of Northern Ireland over the rules that would apply there.
The PM also warned that it risked "weakening the delicate balance" of the Good Friday peace agreement because unionist parties like the DUP are so unhappy with it.
The prime minister called for "flexible and creative solutions" and "alternative arrangements" - based on technology - to avoid a hard border.
He said the backstop should be replaced with a commitment to put in place such arrangements as far as possible before the end of the transition period - currently the end of 2020 under Mrs May's deal.
If they were not in place by the end of the transition period, Mr Johnson said the UK was "ready to look constructively and flexibly at what commitments might help".
"Time is very short. But the UK is ready to move quickly, and, given the degree of common ground already, I hope the EU will be ready to do likewise," he wrote.
"I am equally confident that Parliament would be able to act rapidly if we were able to reach a satisfactory agreement which did not contain the backstop."