In her cautiously well-received Florence speech, May proposed a transition period of around two years after Britain's formal departure date in March 2019, during which it will still effectively follow European Union laws.
'The UK will honour commitments we have made during the period of our membership, but it is obvious that reaching a conclusion on this issue can only be done in the context of and in accordance with a new deep and special partnership with the European Union, ' he said.
And Brussels is now likely to object to Britain's more detailed terms, which suggest that the UK Government would not formally sign off on a financial settlement until after trade talks have at the very least been opened.
"The country needs to decide who is going to lead - the United Kingdom can not trust a government that can not trust itself".
The EU has been insisting their needs to be "sufficient progress" on Britain's outstanding financial commitments to the EU before any new relationship can be discussed.
Meanwhile, moderates with the Tory party, including former prime ministers David Cameron and John Major, and leading figures like George Osborne and Ruth Davidson are said to have gathered behind Rudd, with the home secretary reported to have said she was keen to run.
We need to get him in a studio with Jeremy Kyle. "Time is of the essence", said Matti Maasikas, the Estonian EU affairs minister, whose country holds the rotating presidency of the 28-nation bloc.
The third condition is the fate of Northern Ireland's border with the EU-member Republic of Ireland where both sides are wary of endangering the Good Friday peace process. "We are laying out concrete proposals and there are no excuses for standing in the way of progress", he said.
The talks were delayed by a week in order to await May's cautiously well-received speech, which was hailed by Barnier as a "constructive" step forward.
Amid the negotiations, May will also make her case to EU President Donald Tusk over lunch in London on Tuesday and then to the bloc's other 27 leaders at a dinner in Estonia on Thursday.
Philip Hammond, the chancellor, has raised further doubts about Theresa May's future after failing to support her plans to fight the next election as Conservative leader, describing her prospects post-Brexit as "not an issue for today".
The long-awaited Brexit talks was launched on June 19, almost one year after Britain voted to leave the bloc by a narrow margin on June 23, 2016.