But Mr Coveney said: "This isn't a question of either side wanting to put up borders, but if you have to protect a functioning single market, just the same way Britain wants to protect its own single market, well then you have to understand that if goods move from one customs union to another then there needs to be some checks unless there is some mechanism that is negotiated and put in place that prevents that".
Speaking to ITV's Peston On Sunday, Ms Sturgeon said she could not commit to backing the EU Withdrawal Bill.
She says Britain wants a close relationship after Brexit, but not one modeled on existing trade deals with Norway or Canada.
She maintained that her view of future British-EU ties was credible and expressed confidence that she would reach a good deal, although she appealed for the bloc to be more flexible in its negotiations. "We want competition between us to be fair and open, and we want reliable transparent means of verifying we are meeting our commitments and resolving our disputes", according to May. Big business echoed his words and even some Brexit campaigners praised the prime minister for her pragmatism.
The EU's main negotiator, Michel Barnier, welcomed the "clarity" May offered and her recognition that Britain faces "trade-offs".
May's speech, entitled "Our Future Partnership", was an attempt to settle doubts over how Britain sees its future outside the European Union and its economic architecture and to try to ease frustrations in Brussels over what they say is a lack of detail.
In a speech Friday aimed at answering critics who accuse Britain of failing to grasp the tough realities of leaving the EU, May will call for "the broadest and deepest possible agreement - covering more sectors and cooperating more fully than any free trade agreement anywhere in the world today".
DFS Requests Information on Deutsche, Other Banks' Dealings With Jared Kushner
But Kushner Companies' spokeswoman said they'd received no letter from the state and called the inquiries "harassment exclusively for political reasons".
Brexit Secretary David Davis is concerned that approach might tie parts of Britain's economy to the European Union without having a say, with one Cabinet source telling the Telegraph: "It would make us a rule taker".
On ITV's Peston on Sunday, Scottish first minister Nicola Sturgeon said that May should be given credit for "injecting some realism" into the debate by admitting in her speech on Friday that the United Kingdom would be worse off after Brexit.
The crucial intervention ahead of the next round of talks with Brussels, appeared to unite warring Remain and Leave factions of the Conservative Party.
Meanwhile, Labour former cabinet minister Lord Mandelson dismissed Mrs May's Brexit plan.
But she sought to reassure companies that an alternative system would be set up.
Labour leader Jeremy Corbyn accused Mrs May of failing to bring "real clarity", stressed he was anxious about her admittance that access to European markets would be reduced, and said she "failed to provide solutions" in particular on the Irish border.
But May's overall message was that the European Union and Britain must both focus on the benefits of their future trade relationship.