Mr Barnier is expected to meet Brexit Secretary David Davis in Brussels for the start of a fifth round of negotiations on October 9, but Mr Juncker's comments suggest a breakthrough in those discussions is unlikely.
Visiting troops with French President Emmanuel Macron, May reiterated Britain's position that security cooperation is not up for debate in London's tense divorce negotiations with the European Union as she seeks to improve the mood and unlock the next stage of talks.
Europe is facing growing threats of terrorism, cyber crime, migration and "Russian aggression", UK Prime Minister Theresa May said in a statement ahead of the European Union summit on digital technologies due in Estonia on Friday.
Britain had aimed to make a breakthrough at a summit in Brussels on October 19-20.
"The informal get-together in Tallinn, arranged before a "digital summit" on issues ranging from data and cybersecurity to taxing online businesses, has no set agenda, allowing May another chance to pitch her Brexit ideas". The EU's chief negotiator Michel Barnier insists it is "indispensable" in ensuring the "consistency" of the rights of EU citizens living in the UK.
"I made that speech to give momentum to the talks and I think we have seen that being shown in the talks that have taken place this week, and further progress has been made", May said on Friday morning.
Mr Davis pursued a more combative tone than his boss and his team didn't go far enough to clarify the concessions Mrs May offered in the speech, according to the person, who asked not to be named as the discussions were private. We need to acknowledge to each other they are out of shape...
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It said that her trip was arranged at the invitation of Burmistrov who visited the North in late July. However, the White House said it was "absurd" to suggest the U.S. had declared war.
Still, he hailed the "new dynamic" that May's speech in Italy last week had inspired and the more detailed British proposals that had resulted from it.
European Union bosses say that not enough progress has been made on major issues such as citizens' rights, Northern Ireland and the final figure for the divorce bill.
"The PM pointed to the commitment made in her Florence speech to incorporate the agreement reached on citizens' rights fully into United Kingdom law and make sure the United Kingdom courts can refer directly to it", the spokeswoman said in a statement.
Speaking ahead of the summit, Ms May said: "From terrorism to cyber-crime, illegal migration to Russian aggression, the threats we face as Europeans are increasing in their scale and complexity".
Trade minister Liam Fox has described EEA nationals in the UK as one of the government's "main bargaining chips" in upcoming negotiations, and May has argued that the UK would be left "high and dry" in negotiations by guaranteeing the rights of EEA nationals without receiving similar assurances for UK nationals living in the EU.
Information for this article was contributed by Jill Lawless of The Associated Press.