Boris Johnson has told the EU the backstop plan for the Irish border must be scrapped because it is "unviable" and "anti-democratic".
In a letter to European Council President Donald Tusk, the PM said the backstop - which aims to avoid a hard border - risked undermining the Northern Irish peace process.
If the plan were removed, Mr Johnson claimed a Brexit deal would be passed by Parliament.
Brussels has not yet responded.
However, the EU has consistently insisted the backstop must remain part of the withdrawal agreement and cannot be changed.
In a phone conversation with Mr Johnson on Monday evening, Irish Taoiseach (prime minister) Leo Varadkar reiterated that the agreement - negotiated by former PM Theresa May but rejected by Parliament three times - could not be reopened.
Mr Johnson's four-page letter to Mr Tusk comes ahead of meetings this week with German Chancellor Angela Merkel and French President Emmanuel Macron.
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 arrangement as "inconsistent with the sovereignty of the UK" and insisted it could not form part of a withdrawal agreement.
He 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."