Labour MP Tom Watson has called on Rupert Murdoch to be questioned by police after a covert recording emerged of him appearing to regret the level of help News Corporation gave to the investigation into alleged wrongdoing at its newspapers.

The News Corporation boss describes payments to police and public officials as "the culture of Fleet Street" in a secret recording made during a meeting with staff from the Sun newspaper this March.

In a transcript of the recording, which was obtained by the investigative website ExaroNews and aired by Channel 4 news on Wednesday evening, he appears to regret the extent of News Corporation's internal management and standards committee's (MSC) co-operation with the police.

After a Sun journalist at the meeting appears in the recording to query the MSC handing over documents to the police, Murdoch is heard saying: "… it was a mistake, I think. But, in that atmosphere, at that time, we said, 'Look, we are an open book, we will show you everything'. And the lawyers just got rich going through millions of emails.

"All I can say is, for the last several months, we have told, the MSC has told, and [**** ****], who's a terrific lawyer, has told the police, has said, 'No, no, no – get a court order. Deal with that.'"

Journalists told Murdoch that they felt like scapegoats and he says: "We're talking about payments for news tips from cops: that's been going on a hundred years, absolutely. You didn't instigate it."

Watson told the Guardian: "I think we need to hear from [Metropolitan police comissioner] Bernard Hogan-Howe when the MSC stopped co-operating with the police inquiry."

In one particularly revealing part of the recording, Murdoch makes reference to those who might succeed him as the head of the media empire. A journalist asks what would happen if the 82-year-old Murdoch was no longer around to give them his support. "The decision would be – well, it will either be with my son, Lachlan, or with Robert Thomson [Murdoch's chief executive at News Corp]. And you don't have any worries about either of them," he tells the meeting.

During the tape, Murdoch accuses the Metropolitan police of incompetence and brands the criminal investigations into journalists as a "disgrace", saying: "The idea that the cops then started coming after you, kick you out of bed, and your families, at six in the morning, is unbelievable."

Watson told Channel 4 News: "What he seems to be saying there is that they stopped co-operating with the police before the Sun staff started to rebel. And what I would like to know is what are they sitting on that they've not given the police. And I'm sure that this transcript and this audiotape should be in the hands of the police tomorrow because I hope that they're going to be interviewing Rupert Murdoch about what he did know about criminality in his organisation."

In a statement, News Corp said: "No other company has done as much to identify what went wrong, compensate the victims, and ensure the same mistakes do not happen again. The unprecedented co-operation granted by News Corp was agreed unanimously by senior management and the board, and the MSC continues to co-operate under the supervision of the courts."

It added that Murdoch had shown "understandable empathy" with Sun staff and would "assume they are innocent until and unless proven guilty".

On Wednesday the government said that the all-party royal charter on press regulation cannot be considered by the privy council until September at the earliest.

The culture secretary, Maria Miller, facing the persistent threat of legal action by the newspaper industry, will put the press's alternative version of the charter to a meeting of the privy council next week, the last such meeting until September.

It is expected that Miller will refer the industry press charter to a sub-committee of the privy council, but it is unlikely to progress because, according to culture department sources, the sub-committee will be told that an alternative supported by the three main party leaders is waiting in the wings.

Miller will not sit on the sub-committee herself, in order to underline the independence of the process and to show she is not pre-judging the issue.

She is likely to be expecting to see the privy council reject the industry's charter and endorse the tougher charter proposed by politicians.