In an op-ed for German newspaper Die Welt, the prime minister noted that, from her point of view, London’s initiatives on Brexit were fair, while Brussels' demands were unacceptable.
"There have been arguments made against our proposals that have been at odds with the reality of trade negotiations elsewhere and indeed the current trading relationship between EU member states," May wrote ahead of the EU Summit in the Austrian city of Salzburg.
She pointed out that the European Union insisted that issues concerning the flow of goods and services could not be separated during the talks.
"But no free trade agreement the EU has ever concluded treats goods and services on the same basis. And most of the relevant services for goods are not covered by EU regulation in any case. Our commitments recognize that in future, when UK firms provide services into the EU, they will follow the same rules as firms in that EU Member State – meaning they simply cannot undercut EU service providers," May said.
The UK prime minister also focused on the settlement of Northern Ireland issue.
"Any settlement must respect the Good Friday Agreement that is the basis for Northern Ireland’s constitutional order and peace and stability there, and must be able to command cross-community support. A hard border, either between Northern Ireland and the Republic of Ireland or between Northern Ireland and the rest of the United Kingdom, would not do so," May said.
She noted that in order to avoid a "hard border" between Ireland and Northern Ireland, the United Kingdom and the European Union needed to agree on the "frictionless" movement of goods.
"This is not the same thing as partial participation in the Single Market: British companies would not enjoy the same legal rights, for example," May said.
The prime minister insisted that any demands to introduce checks between mainland Britain and Northern Ireland were unacceptable.
"Neither side can demand the unacceptable of the other, such as an external customs border between different parts of the United Kingdom – which no other country would accept if they were in the same situation, or the UK seeking the rights of EU membership without the obligations," she wrote.
The prime minister concluded by saying she hoped that London and Brussels would manage to find a way to maintain good and friendly relations after the Brexit.
"With goodwill and determination on both sides, we can avoid a disorderly exit and find new ways of working together. In or out of the EU, we are still all part of our European family of nations and must stay good friends who support each other’s safety and prosperity," May added.
Earlier this month, Barnier called the UK government’s proposals on post-Brexit trade relations unacceptable and "dead." He pointed out that the United Kingdom could not have any special preferences in the EU single market. Barnier insisted that the United Kingdom could either stay in the single market and comply with all the relevant trade regulations or leave the trade bloc altogether. Barnier also said that in the event that there is no Brexit deal, the United Kingdom should introduce border checks between mainland Britain and Northern Ireland in order to prevent the emergence of a "hard border" between Northern Ireland and the Republic of Ireland.
In June 2016, the United Kingdom voted to leave the European Union. Although Brexit is scheduled for March 29, 2019, London and Brussels still cannot agree on a number of key issues, including the Irish border and the customs arrangements, making a no-deal scenario a possibility.
On September 10, Barnier said that agreements could be finalized in late October or the beginning of November.