ROME (Reuters) - British Prime Minister Theresa May should make the EU firm offers by next week to break deadlock in Brexit talks if she wants an orderly withdrawal and transition to a new relationship, the bloc’s negotiator said on Thursday.

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Speaking to the Italian parliament on the eve of a speech that May will deliver in Florence, Michel Barnier said he was wondering why there was still “major uncertainty” on all key issues Brussels wanted settled before opening talks on the transition to a future trade deal it expects her to ask for.

“To make progress, we are waiting for clear commitments from the UK on these precise issues,” Barnier said, noting demands for European Union citizens in Britain to have their rights there protected by EU courts and for London to pay a hefty bill before leaving.

“We will listen attentively and constructively to Theresa May’s important speech tomorrow in Florence,” added Barnier, who holds a new round of Brexit negotiations in Brussels next week.

“A rapid agreement on the conditions of the UK’s orderly withdrawal, and a transition period, is possible ... For that to happen, we would like the United Kingdom to put on the table, as soon as next week, proposals to overcome the barriers.”

An agreement on orderly withdrawal was “a precondition for any constructive and trustworthy discussion on our future relationship with the United Kingdom.”

Barnier renewed criticism of Britain’s treatment of 3 million EU expatriates in the country, noting deportation letters sent to some recently in error and allegations that May’s interior ministry defied a High Court order over an asylum case.

London has rejected EU demands that the European Court of Justice remain the ultimate arbiter for EU citizens in Britain and says its own court system offers sufficient guarantees. But Barnier’s position remains that the EU wants its own safeguards.

TWO-YEAR TRANSITION?

The BBC said after May discussed the Florence speech with her cabinet that she would ask for a two-year transition period after Brexit in March 2019, would refer to “paying our dues” and say that a deal with the EU would be unlike other countries’.

European capitals expect May to try to break a deadlock in the talks over the coming days by signaling a readiness to pay the EU to stay in the single market for two to three years after Brexit, easing fears of a disruptive break.

Barnier recalled that the EU was ready to consider a “short transition period if the British government requests one” but also underlined that would entail accepting EU rules, including on its budget. And he noted that the more benefits Britain wanted the more it would have to accept obligations to the EU.

The EU executive estimates Britain may owe something like 60 billion euros ($70 billion) on leaving, while Britain questions whether it has any legal liability. EU officials have said that British payments to the EU during a post-Brexit transition -- its current annual payments are about 10 billion euros -- cannot be offset against a need to settle a pre-Brexit “divorce bill”.

However, they acknowledge that the British government might find ways to present the various financial agreements it makes in a way it hopes may limit criticism from hardline Brexit supporters of any divorce settlement and transition payments.