Speaking on Wednesday, Barnier said that the absence of an agreement between the two sides would have “serious consequences for everyone.”
Barnier, who is leading the EU side in the upcoming Brexit talks, told a meeting, “A 'no deal' scenario is not our scenario.”
He said, “We want an agreement, we want to succeed, not against but together with the UK.”
The French official was addressing a plenary session of the Committee of the Regions (CoR).
He told CoR members that the key objective of the upcoming negotiations will be “guaranteeing the rights of European citizens, in the long-term.”
Barnier, a European Commissioner, said, “Our watchword will be: ‘Citizens first’.”
He was speaking ahead of the UK triggering article 50 of the Lisbon Treaty next week. This will pave the way for formal Brexit talks to start. They are expected to last up to two years.
The EU's chief Brexit negotiator added that the “uncertainty” of local and regional governments delivering EU-funded projects “must also be resolved.”
Noting that these programmes were “approved by 28, financed by 28 and so should benefit” 28 EU members, he said, “When a country leaves the Union, there is no punishment.
“There is no price to pay to leave. But we must settle the accounts. We will not ask the British to pay a single euro for something they have not agreed to as a member.”
Barnier, speaking publicly for the first time about the impact of Brexit on European cities and regions, said, “In the same way, the 27 will also honour their commitments concerning the UK, its citizens, companies and regions.”
Barnier told CoR members, “The first condition is the unity of the 27, which goes hand in hand with transparency and public debate.”
He said that since he had taken up office on 1 October, he had met the governments of all 27 member states and over the past few weeks started a second tour of the capitals to meet the governments again, as well as the national parliaments, trade unions and professional organisations.
“During this period and for the duration of the negotiations, I will naturally work in close cooperation with the Council, the European Parliament and all other bodies and institutions of the EU.”
Barnier told the plenary, “Unity is the first condition for reaching an agreement in the negotiations. It is of course in our interests. But it is also - and I want to say this to our British partners - in the UK's interest. Because, at the end of the day, we will both need - you and us - a united Europe to reach a deal.”
He said, “These negotiations cannot take place in secret. We will negotiate in a transparent and open manner, explaining to everyone what we are doing. During these negotiations, we must also explain objectively what ‘leaving the EU’ means, for the withdrawing country and for the other member states.
“We need to tell the truth to our citizens about what Brexit means. The second condition for reaching an agreement is removing the uncertainty created by the UK's decision to leave the EU.
“This uncertainty is first and foremost that of 4.5 million citizens: the Polish students who have access to British universities under the same conditions as British students and the British pensioners who are resident in Spain and who benefit from healthcare under the same conditions as Spanish pensioners.
“There are also the Romanian nurses and doctors who contribute to the quality of healthcare in the UK. We hear their doubts. We understand their worry, and we must act effectively in response.”
He told the packed meeting, “The issues at play are complex, whether they are residency rights, access to the labour market, pension or social security rights, or access to education.
“We will work methodically on each of these points.
“We will not leave any detail untouched, and we are already working with all member states on this. It will take time, several months certainly. We must do serious legal work on this with the UK.”
Barnier went on, “But we can and we should agree - as soon as possible - on the principles of continuity, reciprocity and non-discrimination so as not to leave these citizens in a situation of uncertainty.”
There was, he noted, yet more uncertainty for regional and local authorities and all beneficiaries of programmes that are currently financed by the European budget.
He asked, “Who are we talking about? It is the beneficiaries of the European social fund, which - with almost €90bn for all regions - helps those men and women who are least qualified and have most difficulty in finding work.”
Uncertainties are also faced by beneficiaries of the European regional development fund, said Barnier.
“We are talking about almost €200bn to support regions in economic difficulty and regions that are isolated and beneficiaries of the Juncker investment plan.”
The plan, he noted, also supports advanced infrastructure in healthcare and energy in the UK.
A third uncertainty created by the UK decision to leave concerns the new borders of the EU, he said, adding, “I think particularly of Ireland. I have been Commissioner in charge of the PEACE programme. I understand the Union's role in strengthening dialogue in Northern Ireland and supporting the Good Friday Agreement, of which the UK is one of the guarantors.
“That is why we will be particularly attentive, in these negotiations, to the consequences of the UK's decision to leave the customs union, and to anything that may, in one way or another, weaken dialogue and peace.”
Barnier told the meeting that each country must honour its commitments to each other.
He said, “If I may quote one of the greatest men of European history, Winston Churchill: ‘the price of greatness is responsibility’. That is true for Britain and for us.”