Mrs May needs the ten DUP votes to help get her Brexit plan through the Commons.
Tory Remainer Dominic Grieve said a border in the Irish Sea would be "unacceptable".
A leaked letter from the Prime Minister to Mrs Foster and her deputy Nigel Dodds set out Mrs May's approach.
May's plan would see the whole United Kingdom effectively agree to remain in the customs union to help avoid a hard border with Ireland as a backstop if no other arrangement can be found.
May's spokesperson told reporters she had said that "while 95% of the withdrawal agreement had been concluded, on the Northern Ireland backstop there are a number of issues that we still need to work through and these are the most difficult".
Mrs May says in the letter that the European Union is still pushing for a "backstop to the backstop".
May's spokesman said ministers agreed at cabinet on the need for an "effective mechanism" within the backstop to ensure "that the United Kingdom can not be held in the arrangement indefinitely".
"It appears the Prime Minister is wedded to the idea of a border down the Irish Sea with Northern Ireland in the European Union single market regulatory regime".
A Downing Street spokesman said: "The Prime Minister's letter sets out her commitment, which she has been absolutely clear about on any number of occasions, to never accepting any circumstances in which the United Kingdom is divided into two customs territories".
However, Brussels has demanded an insurance policy to avoid a hard border in Ireland.
The question of how to avoid a "hard border" between Northern Ireland and the Irish Republic after Brexit has proved to be the toughest issue in the exit talks.
Amid concerns about the UK's ability to exit such a scenario, the prime minister has also come under pressure to publish legal advice on potential post-Brexit arrangements for Northern Ireland.
Downing Street has played down suggestions that a Brexit deal is imminent, after European Council president Donald Tusk appeared to indicate a breakthrough could come within the next week.
Brexiters are adamant that there must be a mechanism that would allow the United Kingdom to leave, and many back one that would allow the United Kingdom to withdraw unilaterally - without Brussels' consent - which has been rejected by figures such as the Irish Prime Minister, Leo Varadkar.
Earlier this week, the prime minister's cabinet discussed a "review mechanism" for bringing the UK's participation in a backstop deal to an end.
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The same warning is provided for the 13-inch non Touch Bar MacBook Pro, which may experience SSD failure and data loss. The iPhone X has problems with its touch screen , Apple has admitted, as it offers customers free repairs.
Rickie Fowler (68) and Tony Finau (65) are at eight under, along with their Ryder Cup captain, Jim Furyk , who had a 65. Kuchar was at 14-under 128 as he tries to end more than four years without a PGA Tour victory.
This includes the original Crackdown , which can be downloaded free of charge in a pre-Black Friday 2018 special. Spotted by Resetera user Neural, Age of Empires has been labeled as available for Xbox One on the Xbox Store .
An Austrian colonel is being investigated on suspicion of spying for Russia , Austrian officials have said. As for the Russians, Foreign Minister Sergei Lavrov said he was "unpleasantly surprised" by the news.
Prior to the arrest, the registered Republican was living in a van plastered with pro-Trump and anti-Democrat stickers. The document said he mailed 16 improvised explosive devices though the US mail to victims across the country.
Known exploits will be dealt with in a future update, though we're not sure if that's at launch or after. Many of these fixes are geared towards PC players, particularly the ultra-wide support and push-to-talk.
Some of those working in the company's Indianapolis office "will stay until March 2019 to publish the remaining program". Reported initially on EGM Now , the news was confirmed by the Dorling Kindersley division of Penguin Random House.
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Australia: Several people stabbed in Melbourne 'terror' attack
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Deschamps Recalls Martial To France Squad
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Could the oil Industry in India See Higher Profits?
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