"It's not a small event, it's a situation that is grave and exceptional", he noted.
Barnier pointed out that a transitional arrangement might be necessary.
It is understood that Brussels wants the scope of initial discussions not to fixate on a number but instead for there to be an understanding on how to calculate the bill, and over what period of time the United Kingdom would settle it.
Winter 2018/19: Once a deal is concluded between the Commission and the United Kingdom, it will go back to the member states of the EU.
Barnier said the first condition for success was the unity of the 27 governments staying in the European Union and the transparency of the negotiations. He noted that Polish students and Romanian nurses in Britain as well as British pensioners in Spain faced great uncertainty over rights to residency and access to the labor market, pensions, social security and education.
"Our main priority for the negotiations must be to create as much certainty and clarity as possible for all citizens, companies and member states that will be negatively affected by Brexit, as well as our important partners and friends around the world".
Theresa May also wants the issue to be dealt with as a priority, while insisting a solution should be reciprocal.
Barnier insisted that Britain would face no punishment for leaving, but added that accounts must be settled.
"We will not ask the British to pay a single euro for something they have not agreed to as a member", he said.
Discussion of the divorce bill is likely to dominate the coming weeks after Mrs May confirmed her intention to initiate Britain's exit next week and then "negotiate hard" for a good deal.
Negotiations must also quickly dispel any uncertainty regarding the EU's external borders after Britain leaves the bloc, in particular the Irish-British land border.
Separately, European Investment Bank (EIB) has estimated that Britain's bill for settling its financial position with the bank could be very costly.
European Union wants divorce deal first, only then talks on future relationship with Britain.
The ability of companies such as easyJet to operate on routes across the European Union has been a major part of their business models, and many have already threatened to concentrate future growth in Europe if the current aviation agreement can't be maintained.
Directly addressing suggestions in the UK that no deal would be better than a bad deal, he issued a warning of the "serious consequences" for Britain leaving without agreement: "the reintroduction of binding customs controls, which will inevitably slow down trade and lead to queues of trucks at Dover; serious disruption to air traffic; an overnight suspension in the movement of nuclear materials to the UK", Barnier said.