News Corporation executives have been considering how to draw a line under the News of the World phone-hacking affair as the Rupert Murdoch-controlled publisher of the tabloid – via its News Group Newspapers subsidiary – braces itself for further celebrity lawsuits in the coming weeks.

This weekend it emerged that former England footballer Paul Gascoigne is planning to sue the paper, claiming his phone was hacked, while others are poised to act after being told that they were referred to in the notebooks of Glenn Mulcaire, the private investigator employed by the tabloid who was jailed for his part in hacking into phones belonging to aides of Princes William and Harry.

The drip-drip of accusations has left News Corp wondering how much more it has to endure, at a time when the company is already quietly softening its legal approach when it is being sued.

Previously News Corp had sought to settle cases, paying Gordon Taylor, head of the Professional Footballers' Association, and publicist Max Clifford, about £1m each.

Now it is letting the cases run, partly to see what evidence there is of hacking by NoW reporters, and also because it does not want to be treated as a "piggy bank" by high-profile claimants.

The company had said that hacking was the work of a single rogue reporter – former royal editor Clive Goodman – who was jailed over the royal voicemail hack along with Mulcaire back in 2007. But News Corp now wants to see what evidence there is of hacking from elsewhere.

The string of civil cases is fuelled by extracts from Mulcaire's notebooks. His notes were seized by the Metropolitan police in the course of their original investigation, and are now gradually being made available to celebrity claimants.

News Corp says it is unaware of the contents of the notebooks, and only gets to see the extracts when the Met releases them to those suing the company.

Last month, the publisher suspended Ian Edmondson, the assistant editor (news) at NoW, after lawyers for Sienna Miller filed a high court claim that alleged he commissioned Mulcaire to hack into the phone of the actor and her friends. Mulcaire's notes relating to Miller showed the name "Ian" written on them.

Charlotte Harris, the lawyer representing sports agent Sky Andrew, said: "It is shocking that it has been left to claimants to push matters through the civil courts in the face of relentless denials from the News of the World that the phone hacking was the work of a rogue reporter. The numerous new claims to be issued shortly suggest a very different story."

The political pressure is likely to intensify on Tuesday when Labour MP Tom Watson asks the government's top legal adviser, attorney general Dominic Grieve, to make a statement to parliament about his department's handling of the phone-hacking affair.

Watson is expected to use a supplementary question to demand that the government forces the Met to reveal how many MPs have been the victim of phone hacking by Mulcaire. Scotland Yard has been slow to notify the majority of victims, and many have only been told after making approaches to the police. Others have contacted their mobile-phone companies in an attempt to confirm suspicions that their voicemail messages were accessed by third parties.

Watson said: "I expect the attorney general to give an unequivocal commitment that his review will ensure that everyone in the Mulcaire evidence file will be informed they were targets of News International."