Pressure lifted on the embattled leader after she struck a deal with the European Union over Britain's divorce terms last Friday, enabling talks to turn to the country's future trading relationship after months of fraught negotiations.
May, weakened after losing her Conservative Party's majority in a June election, has so far carried her divided government and party with her as she negotiated the first phase of talks on how much Britain should pay to leave the European Union, the border with Ireland and the status of European Union citizens in Britain.
All stressed that the second round of talks would be hard.
Mr Tusk said on "EU leaders agree to move on to the second phase of #Brexit talks".
"There cannot be any border controls between Northern and southern Ireland, there cannot be border controls between Northern Ireland and the United Kingdom, but there can between United Kingdom and the European Union", he said.
British Chancellor of the Exchequer Philip Hammond said Saturday the U.K.'s trading relationship with the European Union would remain largely unchanged after Brexit. So nervous are the markets that his comments led to a fall in the pound against the dollar and the euro.
It would see the United Kingdom and EU continuing the regulatory alignment that exists today, and the formation of a new customs union similar to the existing one.
The EU also wants more clarity from the United Kingdom on what sort of trade and future relationship it envisages after Brexit to be able to start discussions on that future after March, when EU leaders adopt another set of guidelines for those talks.
The guidelines also stipulate that the United Kingdom must adopt all new European Union laws created during the transition period, while being excluded from any decision-making role.
On Monday, Ms May will tell MPs: "We will prepare for our future independent trade policy by negotiating - and where possible signing - trade deals with third countries, which could come into force after the conclusion of the implementation period".
She added: "The British people voted for control of their borders, their laws and their money". However, he added, in a masterpiece of understatement, "There does seem to be quite diverse opinions as to what that should look like".
The deal nearly fell apart last week, when May's Northern Irish allies rejected an initial agreement for fear that a promise to protect a free border with European Union member Ireland could separate their region from the rest of the UK.
Collectively this means that the United Kingdom will continue to be a member of the Single European Market and Customs Union and accept EU immigration and legal jurisdiction while not even having discussions on a post-Brexit trade deal for months to come.
The document issued, and statements made repeatedly by Juncker, will act as a spur to the pro-Remain wing of the Tories.