The European Union's chief negotiator said the Prime Minister's call for a two-year transition period and commitment to maintain some payments to Brussels showed a "willingness" to move forward.
But he also made clear that any future relationship would have to be based on a balance of "rights and obligations" and would have to maintain the EU's legal order - a reference to Ms May's ongoing determination to end the European Court's remit in the UK.
In a statement released shortly after the end of Ms May's 35 minute speech in Italy, he said the intervention "expressed a constructive spirit which is also the spirit of the European Union during this unique negotiation".
He went on: "The speech shows a willingness to move forward, as time is of the essence.
"We need to reach an agreement by autumn 2018 on the conditions of the United Kingdom's orderly withdrawal from the European Union. The UK will become a third country on 30 March 2019."
He told Ms May that the "sooner we reach an agreement on the principles of the orderly withdrawal" the sooner the EU would ready to discuss a future trading relationship - the key objective of Ms May's Brexit strategy.
Mr Barnier noted that for the first time the UK had requested to stay in the single market and adhere to its rules - including on free movement and accepting the rule of EU law - during a two year transition period. He said the EU would be able to consider "this new request".
On the future relationship he said: "The EU shares the goal of establishing an ambitious partnership for the future.
"The fact that the government of the United Kingdom recognises that leaving the European Union means that it cannot keep all the benefits of membership with fewer obligations than the other Member States is welcome.
"In any case, the future relationship will need to be based on a balance of rights and obligations. It will need to respect the integrity of the Union's legal order and the autonomy of its decision-making."
Mr Barnier will meet the UK's negotiators including Brexit Secretary David Davis on Monday for the fourth round of Brexit talks.
He said he looked forward to Mr Davis "explaining the concrete implications" of the Prime Minister's speech.
He went on: "Our ambition is to find a rapid agreement on the conditions of the United Kingdom's orderly withdrawal, as well as on a possible transition period."