Gerry McCann, the father of missing Madeleine, has told ITV News he would have liked Lord Justice Leveson's report "to have gone further" but added that he would be "disappointed" if the Prime Minister did not implement the recommendations in full.
Mr McCann told Paul Davies that MPs should be "given a free vote" and urged them to "do the right thing".
A man who won libel damages after his arrest over the murder of a woman has written to MPs asking them to back Leveson's recommendations.
About 2,000 editors will sign up to a new independent press watchdog, the chairman of the Press Complaints Commission has said.
The Prime Minister has taken a political gamble by holding differing views to the victims, the Leveson inquiry and a majority of MPs.