Letwin wouldn't specify the model being looked at, although some present speculated that it involved a member of the judiciary or a Parliamentary office.
The Prime Minister warned editors that unless the Leveson proposals were accepted in entirety he would introduce a press law.
Newspaper editors will meet again tomorrow and have been told to report back to the Culture Secretary Maria Miller on Saturday with a "timetable for progress". The Government is hopeful that draft proposals for a new independent regulator would be in place by Christmas.
The notion of a "verifier" without powers enshrined in law was quickly rejected by press reformers.
The idea was "incoherent," said Evan Harris, associate director of pressure group Hacked Off. "If Leveson's recommendations concluded there must be a statute to create an independent verifier of the self-regulation system it is gibberish to say you are going to implement Leveson in full without that."
Some victims of phone hacking had gathered at Downing Street to heckle the editors, who knew that they had to put aside intense industry rivalries and present a unified front in order to head off the introduction of a press law.
Referencing the film The Godfather, Tony Gallagher, editor of the Daily Telegraph, tweeted that the meeting "felt like the summoning of the five families".
Those present included Lionel Barber, editor of the Financial Times, James Harding, editor of the Times, Chris Blackhurst, editor of the Independent, Alan Rusbridger, editor of the Guardian, Geordie Greig, editor of the Mail on Sunday, Dominic Mohan, editor of the Sun and John Witherow, editor of the Sunday Times.
Paul Dacre, editor-in-chief of the Daily Mail, was a notable absentee - unable to attend for personal reasons.
There was only one woman among the journalists - Sarah Sands, editor of the London Evening Standard.
The Prime Minister spent 20 minutes in the meeting before handing over to Letwin who stressed the urgency of implementing the reforms and the importance that Leveson's proposals were accepted "line by line". The Deputy Prime Minister Nick Clegg has backed Leveson's call for statutory underpinning.
Lloyd Embley, editor-in-chief of the Daily Mirror, tweeted: "There is a firm belief that papers can deliver Leveson principles far more quickly without legislation - better for public and free speech.
- Independent