The two leaders met for a morning discussion on the fringes of an EU summit in Estonia.
It was the first meeting between the pair since Mrs May’s Florence speech in which she held out an olive branch to the bloc by offering to continue to pay budget contributions.
She was hoping to persuade the German chancellor, fresh from a disappointing domestic election result, that trade talks should now begin.
Number 10 said Mrs Merkel had welcomed the Florence speech and spoke of “good progress” made in the latest round of negotiations.
A Downing Street spokesman said: “The Prime Minister and the Chancellor both agreed on the importance of settling the issue of citizens’ rights at the earliest opportunity.
“The PM pointed to the commitment made in her Florence speech to incorporate the agreement reached on citizens’ right fully into UK law and make sure the UK courts can refer directly to it.
“The PM also stressed it was in everybody’s interest to agree to a time-limited implementation period once Britain leaves the EU, to provide certainty to businesses and others in both Britain and the EU.”
The Prime Minister is also expected to have talks on the sidelines in Tallinn with several other leaders as she presses her case.
The meetings could anger European Union chiefs who have repeatedly demanded all Brexit negotiations go through Brussels rather than be conducted between UK officials and head of states.
During informal talks, she will also tell her European counterparts that Britain is unconditionally committed to the defence and security of the continent.
She has told European counterparts the UK’s role in Europe’s defence has “never been more vital” and point out Britain has the largest defence budget, top-class security and intelligence services, and is a key player in Nato.
Mrs May said: “As we prepare for Brexit, I want to build a bold, new security partnership with the EU.
“A partnership that reflects our shared history, promotes our common values and maintains a secure and prosperous Europe.”
Britain, which had been accused of using security as a bargaining chip, is seeking a bespoke deal on links with the EU under proposals to maintain co-operation on fighting terrorism and serious crime after Brexit.
Last night, the Prime Minister attended a working dinner with the other EU leaders ahead of the Tallinn summit, which is billed as a discussion on innovation and research.
Mrs May’s latest intervention comes after high-profile meeting with EU Council President Donald Tusk and Irish Prime Minister Leo Varadkar at Downing Street earlier this week.
it follows an announcement by David Davis and Michel Barnier yesterday that the two sides had made significant progress in negotiations.
The chief EU negotiator praised a “new dynamic” to Brexit negotiations created by concessions Mrs May’s speech.
But he insisted progress still needed to be made on EU citizens’ rights, the Irish border and a financial settlement before talks could move on to a transition period and trade.
Mr Barnier said: ”I think it is positive that Theresa May’s speech made it possible to unblock the situation to some extent and give a new dynamic to the situation.
“But we are far from being at the state and it will take weeks or maybe even months where we will be able to say, ‘Yes, there has been sufficient progress on the principles of this orderly withdrawal’.”
Mr Barnier said he could not even discuss Mrs May’s proposal for a period of transition until there had been sufficient progress on the issues of the UK’s “orderly withdrawal”.
On the rights of EU citizens living in the UK, he welcomed Britain’s assurance that it would adopt “EU law concepts” but insisted they had to be enforced through the European Court of Justice.
Mr Davis said that there had been some “decisive steps” forward, and called for “pragmatism” in resolving the outstanding issues.
He said: ”We have made important progress and capitalised on the momentum created by the Prime Minister’s speech.
“We are working quickly through a number of complex issues but there remain some points where further discussion and pragmatism will be required to reach an agreement.”