Barnier said the speech showed a willingness to move forward, "as time is of the essence."
Barnier's comments came ahead of his latest meeting wth David Davis, his counterpart on the UK side.
The two will meet on Monday to begin the fourth round of the negotiations. The Frenchman will also have a discussion with Parliament's Brexit steering group, as well as with all member states in the General Affairs Council (GAC).
In her speech, May suggested that the Brexit process might be extended by two years, from 2019 to 2021.
Barnier responded by saying, "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 added, "Our priority is to protect the rights of citizens. EU27 citizens in the UK must have the same rights as British citizens today in the EU. These rights must be implemented effectively and safeguarded in the same way in the UK as in the EU, as recalled by the European Council and European Parliament.
"May's statements are a step forward but they must now be translated into a precise negotiating position of the UK government."
"With regard to Ireland, the UK is the co-guarantor of the Good Friday Agreement."
He said, "The speech does not clarify how the UK intends to honour its special responsibility for the consequences of its withdrawal for Ireland. Our objective is to preserve the Good Friday Agreement in all its dimensions, as well as the integrity of the single market and the customs union.
"The UK recognises that no member state will have to pay more or receive less because of Brexit. We stand ready to discuss the concrete implications of this pledge. We shall assess, on the basis of the commitments taken by the 28 member states."
He went on, "For the first time, the UK government has requested to continue to benefit from access to the single market, on current terms, and to continue to benefit from existing cooperation in security. This is for a limited period of up to two years, beyond its withdrawal date, and therefore beyond its departure from the EU institutions."
Barnier said, "The EU shares the goal of establishing an ambitious partnership for the future. The fact that the government of the UK recognises that leaving the EU 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."
He added, "The EU will continue to insist on sufficient progress in the key areas of the orderly withdrawal of the UK before opening discussions on the future relationship. Agreeing on the essential principles in these areas will create the trust that is needed for us to build a future relationship together."
Barnier noted,"Our ambition is to find a rapid agreement on the conditions of the UK's orderly withdrawal, as well as on a possible transition period."