A well-placed source felt the majority of Conservative potential rebel MPs, such as Ken Clarke, who have spoken in favour of going ahead with stage two of the Leveson Inquiry into press ethics, are now more likely to back down in an imminent Commons vote on whether it should proceed.
The source said the Government is also “very confident” of defeating an attempt to force newspapers who are taken to court and are not sanctioned by a state-backed regulator paying the costs of both sides regardless of whether they win, which had the backing of Tory MP Bill Wiggin.
Prime Minister Theresa May will seek to overturn attempts by the House of Lords to insert Leveson 2 and the so-called "Section 40" costs provision into the Data Protection Bill, which is expected to return to the Commons in the coming weeks.
The peers are backed by Labour and there have been suggestions that Tory rebel MPs could support them in crunch votes, although it is unclear whether there would be enough rebels for Jeremy Corbyn's party to win.
But after the Independent Press Standards Organisation (IPSO), the independent regulator backed by most of the press, announced the creation of a low-cost compulsory arbitration scheme, the Tory source felt “that today’s decision by IPSO would satisfy the vast majority of Conservative MPs who have concerns over the Leveson elements of the Data Protection Bill.”
The source also said the Government is “very confident in defeating Labour’s attempts to introduce the Section 40 amendment, which will see newspapers who are taken to court paying the costs of both parties regardless of who won the action.”
The deadline for tabling fresh amendments to the Bill passes on Thursday.
IPSO's announcement will mean that someone who has a genuine claim against a newspaper who could have gone to court, for example for libel, invasion of privacy, data protection or harassment, can ask for arbitration of their claim at a maximum cost of Â£100, and the newspaper cannot refuse.
It will also include a higher level of damages, with arbitrators empowered to award claimants up to Â£60,000, including aggravated damages.
Matt Tee, IPSO chief executive, said: "Lord Justice Leveson stressed the importance of having a low-cost means of people that had been wronged by a newspaper getting compensation, without the expense of court and legal fees. The new IPSO scheme does exactly that and the papers are not able to choose which cases they take.
"Once IPSO has accepted an arbitration claim, a senior barrister is appointed as the arbitrator. He or she will make a preliminary ruling on some core issues in the dispute. This provides a basis for settlement talks and may indicate the likely success of the claim. If one of the parties wants to continue the claim, it goes to a final ruling, after which the arbitrator can award damages and fees, if the claim is successful."