Boris Johnson says he is willing to give the EU the "commitments" it needs that alternatives to the Northern Ireland backstop can be put into effect.
Striking a new tone in the Brexit negotiations, the prime minister's letter to EU Council President Donald Tusk will be seen as an attempt to convince Europe's leaders that he is willing to provide guarantees that other arrangements will be viable.
He wrote: "The backstop should be replaced with a commitment to put in place [alternative arrangements] as far as possible before the end of the transition period, as part of the future relationship.
"I recognise that there will need to be a degree of confidence about what would happen if these arrangements were not all fully in place at the end of [the transition] period.
"We are ready to look constructively and flexibly at what commitments might help."
"Alternative arrangements" to the backstop have long been touted by supporters of Brexit as a potential solution to the Northern Ireland border problem. They include livestock checks away from the border and a "trusted trader" programme for goods.
Mr Johnson also reiterated his opposition to the backstop - which would force the UK into a customs union with the EU if the border between Ireland and Northern Ireland could not be kept open.
He wrote that the backstop is "simply unviable" and "anti-democratic and inconsistent with the sovereignty of the UK as a state".
In the same letter, he challenged the EU to make a legal promise not to impose a hard border. The prime minister wrote: "We would be happy to accept a legally binding commitment to this effect and hope the EU would do likewise."
Details of the letter to Mr Tusk emerged as a senior US politician warned that legislators would move to block a future US-UK trade deal if it puts the Good Friday Agreement at risk by introducing a hard border between Northern Ireland and the Republic of Ireland.
US Senate Democratic leader Chuck Schumer has written to US secretary of state Mike Pompeo warning that Congress could work on a cross-party basis to block a deal.
Mr Schumer called for the Trump administration to stop "over-promising an unconditional and unrealistic" post-Brexit trade agreement with the UK. The letter was also sent to Foreign Secretary Dominic Raab.
Mr Schumer's intervention coincided with President Trump tweeting details of a phone call with Mr Johnson on Monday about a future trading relationship.
He tweeted: "Great discussion with Prime Minister @BorisJohnson today.
"We talked about Brexit and how we can move rapidly on a US-UK free trade deal.
"I look forward to meeting with Boris this weekend, at the @G7, in France!"
A Downing Street spokesperson said: "The prime minister spoke to President Trump today, ahead of the G7 Summit in Biarritz.
"They discussed economic issues and our trading relationship, and the Prime Minister updated the President on Brexit. The leaders looked forward to seeing each other at the Summit this weekend."