The Prime Minister is likely to face a major backlash from Eurosceptics within her party over the plans, with Leave-backing Cabinet members expected to resist such concessions.
British and European Union negotiators are understood to have agreed in principle to an all-UK backstop plan to avoid a hard border in Northern Ireland that would remove the final major obstacle blocking a withdrawal agreement.
The EU chief negotiator accepted new administrative procedures for goods travelling between Northern Ireland and mainland Britain would be "politically sensitive" but he noted: "Brexit was not our choice; it is the choice of the United Kingdom".
Former Conservative leader and Brexit supporter Iain Duncan Smith warned the prime minister she should "listen very carefully" to the DUP, which he said echoed many Tory concerns.
Suspicions remain among hardline Tory Brexiteers that Mrs May is heading for a compromise which could tie the United Kingdom to European Union customs arrangements indefinitely - something which Boris Johnson has warned would reduce Britain to a "permanent European Union colony". It would have implications not just for Brexit legislation - 50% of which would not have passed without DUP support - but also for the Budget, welfare reform and other domestic legislation.
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According to spokesman Margaritis Schinas, Barnier "recalled that decisive progress must be made in time for the October European Council next week and negotiations at technical level will continue this week".
The DUP has 10 MPs, although Ian Paisley can not vote in the Commons until November 20 after being suspended for failing to declare two family holidays paid for by the Sri Lankan government. It must be one of the worrying questions for Downing Street.
But he stuck to the EU's rejection of London's plan for a "regulatory framework for goods", saying it would give Britain an unfair competitive advantage by allowing access to parts of the bloc's single market without ensuring the country honours all of its conditions. The Government has said it will publish revised proposals for the Irish border "in due course".
A meeting between the UK Government and devolved administration was meanwhile witness to "heated exchanges" over Brexit and migration policy.
Blair, a prominent voice opposed to Brexit, resigned as prime minister in 2008.