A well-placed Tory source felt the majority of Conservative potential rebel MPs such as Ken Clarke, who have spoken in favour of the second stage of the Leveson Inquiry into press ethics, are now more likely to back down in an imminent Commons vote on whether it should go ahead.
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.
Prime Minister Theresa May will seek to overturn attempts by the House of Lords to insert Leveson 2 and Section 40 into the Data Protection Bill when it returns to the Commons in the coming weeks.
They are backed by Labour and there have been suggestions that rebel Tory MPs could also support them in crunch votes on the Bill.
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 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.”
They also said they are “very confident in defeating Labour’s attempts to introduce 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 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 £100, and the newspaper cannot refuse.
It will also include a higher level of damages, with arbitrators empowered to award claimaints up to £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."