May, battling to strike a deal with Europe that appeases both sides of her deeply divided party, used a speech in London on Friday to warn that the country needed to face up to some "hard facts" that "neither of us can have exactly what we want".
May conceded in her speech that access to markets would be "less than it is now".
"There are inevitably a few small points that will concern Leave campaigners but we must all recognise that everyone will have to give up something to get a deal, so now is not the time to nitpick", he said.
"We are leaving the single market".
While she will say that she wants Britain to have the freedom to strike trade deals around the world, it should be possible to agree a relationship with the European Union where they continue to "support each other's interests".
Mrs Batters said high quality, safe and affordable British food is a "public right" and needs to be placed at the centre of Brexit negotiations.
Across the English Channel, E.U. leaders were listening closely, hoping for concrete solutions to issues such as the Northern Ireland border and how to achieve the "as frictionless as possible" trade that May has demanded.
In what may be seen as a swipe at United States president Donald Trump a day after he announced new tariffs on steel, Mrs May said: "In all these areas, bold and creative thinking can deliver new agreements that are in the very best interests of all our people - both in the UK and across the EU".
Meanwhile, Mr Barnier welcomed the speech for offering "clarity" on the UK's aims and a recognition there would be "trade-offs".
Reacting, Leo Varadkar said: "I remain concerned that some of the constraints of leaving the Customs Union and the Single Market are still not fully recognised".
He told Today: "We hoped maybe that in some aspect the United Kingdom could reconsider (single market and customs union membership) because that could open a lot of doors for solutions, for example for the Irish/Northern Irish border".
Mrs May said that in chemicals, aviation and medicine, Britain would seek to abide by European Union regulation, and she proposed a streamlined Customs partnership with the same tariffs at the border for goods sold to the EU. She didn't outline specifics on what such an arrangement would look like.
Alliance deputy leader Stephen Farry said Mrs May's Brexit speech did "little to address the many contradictions in the UK's Brexit position".
She again ruled out creating a border in the Irish Sea as it would "damage the integrity of our precious Union".
"May was preparing the ground for medicine that both factions in the Conservative Party's heated debate on Europe will find hard to swallow", he wrote.
The speech comes at the end of a rough week for May.
With (Other OTC: WWTH - news) an Italian election and a vote on a German coalition due on Sunday, investors have also been reluctant to take on any new large positions.
Alternatively, the prime minister said, there could be a technological solution that could make customs checks unnecessary on the island of Ireland and elsewhere. If Brexit flops, he said, "there will be the most bad backlash".
British Prime Minister Theresa May on Wednesday rejected any plans that would mean a Brexit border between Northern Ireland and neighbouring Irish Republic.
In a high-profile speech at London's Mansion House, the Prime Minister set out five "foundations" to underpin the new relationship - including binding commitments to fair competition and a new arbitration mechanism to resolve disputes. The venue was hastily relocated from the northeastern city of Newcastle, as Britain struggles to cope with its heaviest snowfall in decades.