Editor Chris Blackhurst has said there is "movement" across the industry towards an agreement on making the new body fully independent of serving editors.
Editors speaking include James Harding of The Times, Alan Rusbridger of the Guardian, Chris Blackhurst of the Independent, and the former News of the World editor Colin Myler.
On Wednesday, speaking on BBC Radio 4's Media Show, Independent editor Chris Blackhurst said the editors had been through all of Lord Leveson's proposals point by point "and pretty much agreed to them all".