Lobbying group Hacked Off indicated it likely would back the deal, saying victims of the scandal likely would accept the deal, The Guardian reported.
"The victims are prepared to accept this compromise," said Evan Harris, the former Liberal Democrat member of Parliament and Hacked Off associate director. "The fact there is all-party agreement means the press are much more likely to fulfill their side of what they have to do, which is to set up a recognized regulator and then to join it, but there is still matters of detail to sort out,"
Several government sources told The Guardian that liberals and conservatives both gave ground.
Speaking on BBC Radio, Harriet Harman, Labor Party's deputy leader.said, "The [regulatory] framework is set up in royal charter, not by statute ... ."
To ensure the royal charter, which is run by the privy council, can't be changes, "there's just a small piece of legislation in the House of Lords today which will say you can't tamper or water down this charter," Harman said.
The language, which would be an amendment to a bill, she said, "has a legal underpinning effect because what we don't want is to have a situation where everybody agrees what the rules should be and then the press lean on ministers and ministers water it down".
Also to assuage media concerns, the amendment would not specifically "mention this charter," so there would be no press regulation by law, Harman said.
Free-press advocates urged lawmakers to vote against the measure, The Sun reported.
"Popular newspapers are bold defenders of the public interest. It protects our liberties and holds power to account," said Tim Luckhurst, a journalism professor at Kent University. "MPs [Members of Parliament] should search their conscience and vote for freedom of expression unlimited by state intervention."
2014: The Year in Music [PHOTOS]