The media lawyer Mark Lewis, who represents the family of murdered schoolgirl Milly Dowler, said he and the Dowlers are "cautiously optimistic" about Lord Justice Leveson's report.
He said he hoped that the MPs who signed an open letter opposing the statutory underpinning of the press regulator would look again at the proposal.
More top news
If the UK position is rejected, the Government could be faced with a choice between remaining in the Customs Union or accepting a hard Irish
The size of the proposed penalty has not been disclosed.
Walter Leroy Moody became the oldest US inmate put to death since executions resumed in the US in the 1970s.