Asked whether she believed Mr Trump was likely to be fully engaged at the G7 given his major meeting with Kim Jong Un in Singapore on Tuesday, the prime minister said: "President Trump was present at the last G7 in Italy, and he came around the table, we agreed a whole variety of issues there".
Brussels had demanded that Northern Ireland remain aligned with European Union customs regulations once Britain leaves, in a bid to prevent a hard border appearing between Northern Ireland and the Republic.
"A backstop that is temporary is not a backstop, unless the definitive arrangement is the same as the backstop", he said. "The only way we can win a vote is if we all vote together at the same time; that's the only way we can defeat the government". "We are happy and ready to step up the frequency of our negotiations".
But the wording of the proposal said only that the United Kingdom "expects" a replacement customs arrangement in place by the end of 2021.
Less than half of all people in Northern Ireland consider themselves British, according to a new survey. Does it respect the integrity of the single market/customs union?
And he questioned whether Mrs May's proposals would deliver an "all-weather backstop" fit for all circumstances.
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"Does the temporary nature of the customs arrangement mean that several adaptations will now be needed?"
The UK government concedes, however, that to uphold its commitments with regard to the Irish border "an approach on regulatory standards ... will also need to be addressed".
Jon Thompson, permanent secretary at Her Majesty's Revenue and Customs, told lawmakers that leaving the European Union with no deal would cost business a similar amount to a customs arrangement known as "max fac" - or maximum facilitation - because companies would have to fill in customs declarations.
In a coded warning to pro-EU Labour MPs, Starmer said there were "very divided views" in the party about a Norway-style model and he was "injecting some honesty about where we are in the party". "How does that fit with the requirements of our Value-Added Tax system?" But EU leaders warned that imposing a time limit would undermine its value.
The government later published its proposal for a backstop plan to avoid a hard border on the island of Ireland, with an ambition for it to end by the time of the next general election. But it also risks causing more domestic trouble for May, who needs to keep her divided Cabinet together and the Northern Irish party that props up her government onside.
Nevertheless, Barnier said he was not rejecting the proposal outright.
Barnier said that while some progress has been made, several issues remain to be resolved before the 28-29 June EU summit. We will not accept such a proposal.