Delivering a speech in Florence, May said there would also be a continuation of European Union free movement rules during this "implementation period", although EU citizens would be required to register.
Preparations necessary for Britain to adjust to a new relationship with the EU after Brexit "point to an implementation period of around two years," she said.
On payments, the British leader added: "I do not want our partners to fear that they will need to pay more or receive less over the remainder of the current budget plan as a result of our decision to leave.
Britain voted to leave the European Union in a referendum last year and is expected to leave the bloc in March 2019.
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The current EU budget goes until 2020 and Britain pays around 10 billion euros into the budget every year.
May also made an apparent concession on the rights of EU nationals living in Britain after Brexit, which EU officials have said should come under the jurisdiction of the Court of Justice of the European Union.
"I want to incorporate our agreement fully into UK law and make sure the UK courts can refer directly to it," she said, referring to the Brexit deal.
May also said that it was "in all of our interests for our negotiations to succeed".
"If we were to fail, or be divided, the only beneficiaries would be those who reject our values and oppose our interests," she said.