The EU parliament's Civil Liberties Committee earlier approved the deal, which will now go before the parliament early next year with approval expected.
The British Home Secretary said the proposal was "an important step and an example of the co-operation we have already seen between the EU and member states to ensure we can share intelligence and information that is going to help us in this battle against the terrorists".
Momentum for the proposal has grown following the terror attack, inspired by the Islamic State (IS) jihadist group, in Paris that killed 130 people.
The European Union began discussions on a Passenger Name Record (PNR) system in 2010 but misgivings about the use and security of personal data held up progress.
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Many members of the European Parliament were doubly suspicious after revelations about mass US intelligence snooping.
"We cannot wait any longer to put this system in place," said Timothy Kirkhope, the British Conservative MEP who is steering the legislation through parliament.
May made her comments at a London conference where six major EU states and the United States agreed "resolute" action to tackle the threat of IS, also known as Daesh.
"We are united in our determination to combat terrorism through a strong yet proportionate national and international response to tackling the threat posed by Daesh and countering violent extremism and radicalisation."
The countries agreed to push for communications service providers to step up measures against content that encourages extremism and improve airport security.