Britain and the European Union have achieved sufficient progress in negotiations on the terms of their separation to begin the next phase of Brexit talks, according to the European Commission President Jean-Claude Juncker and British Prime Minister Theresa May.

Britain estimates the ultimate cost of meeting its financial obligations to the European Union on Brexit at 40 to 45 billion euros ($A62 to $70.3 billion), a senior UK source says.

“We expect the range to be between 35 and 39 billion pounds,” the spokesman said. “We would look at it as a fair settlement of our obligations.”

Mr Juncker will recommend that negotiations now move to defining future cross-Channel relations and trade.

“I believe that we have now made the breakthrough that we needed,” he told a morning press conference with British Prime Minister Theresa May at his side.

Ms May hailed intensive talks over the past few days, saying all parties’ efforts had delivered “a hard-won agreement in all our interests”.

“It is crucial for us all that we continue working closely together on issues such as trade, research, security and others,” Mr Juncker said.

“We will take things one step at a time, starting with next week’s European Council, but today I am hopeful that we are all moving towards the second phase of these challenging negotiations and we can do this jointly on the basis of renewed trust, determination and with the perspective of a renewed friendship.”

The EU’s chief Brexit negotiator Michael Barnier lauded the accord as “the first hurdle”, but he added there were “more to come”.

Ms May confirmed speculation the agreement would guarantee the rights of three million EU citizens in the UK, their residency “enshrined in UK law and enforced by British courts”.

There will be “no hard border” between Northern Ireland and the Irish Republic, Ms May continued, thus preserving the “constitutional and economic integrity of the United Kingdom”.

“This government will continue to govern in the interests of the whole community in Northern Ireland and uphold the agreements that have underpinned the huge progress that has been made over the past two decades,” she said in a statement.

Foreign Secretary Boris Johnson, often seen as PM-in-waiting, is calling for more progress in the talks.

Irish Foreign Minister Simon Coveney said the agreement fully guaranteed there would be no hard border on the island of Ireland once Britain left the EU.

Observers believe EU leaders meeting in Brussels next week will likely endorse the assessment that enough progress has been made on the terms of Britain’s financial settlement, the status of Irish borders and the rights of citizens hit by Brexit.

While Mr Juncker expressed cautious optimism about the talks’ progress, British Foreign Secretary Boris Johnson, the man often mooted as a potential replacement for Ms May at 10 Downing Street, was keen to see even more progress, and sooner.

“Whatever way we devise for getting onto the body of the (Brexit) talks, it’s got to be consistent with the whole of the United Kingdom taking back control of our laws, of our borders and of our cash,” Mr Johnson said.

– with ABC and wires