A group of high-profile senior MPs have claimed that Ofcom made a “serious legal error” in its handling of Rupert Murdoch’s bid for Sky and could face a judicial review unless the government orders a full investigation into the threat to broadcasting standards from the deal.

Ed Miliband, the former Labour leader, Sir Vince Cable, the Liberal Democrat leadership candidate, Kenneth Clarke, the former Conservative chancellor, and Lord Falconer, the Labour peer and former lord chancellor, claim in a letter to the government that there was a “fundamental failure” in Ofcom’s investigation into the proposed £11.7bn takeover of Sky by 21st Century Fox. Fox is controlled by Rupert Murdoch and his sons, Lachlan and James.

The MPs say that the media regulator did not properly consider allegations of sexual harassment at Fox News and the phone-hacking scandal at the News of the World as part of a review into whether a combined Fox and Sky would be committed to meeting broadcasting standards.

When these factors are taken into consideration against the correct legal threshold, then the bar is met for a detailed investigation into broadcasting standards by the Competition and Markets Authority (CMA), they add.

Miliband and Cable are longstanding critics of the Murdochs and the prospect of them controlling Sky. Cable, the former business secretary, was forced to hand over responsibility for assessing the Murdochs’ failed 2010 bid for Sky after telling undercover Daily Telegraph reporters that he was at war with the media mogul.

The MPs have written to Karen Bradley, the culture secretary, as part of a government consultation about the deal, which closed on Friday.

Bradley said last month that she was minded to refer the deal to the CMA for a six-month examination on the basis of media plurality. This was after Ofcom concluded in a report that the Murdoch family could have an increased influence over news and politics in the UK if Fox bought the 61% of Sky that it does not already own.

However, Bradley said she did not plan to also refer the deal to the CMA on the grounds of broadcasting standards. Ofcom concluded there were “no broadcasting concerns” and in a separate report cleared the Murdochs as “fit and proper” owners of Sky, despite the sex harassment scandal at Fox News representing a “significant corporate failure”.

Bradley is expected to announce a final decision on whether to refer the deal to the CMA as early as next week.

The MPs claim in their letter that Ofcom “failed in its duty as a regulator” because it did not consider the “damning evidence” against the Murdochs in relation to the correct threshold for launching a broadcasting standards probe. The threshold is lower than Ofcom recognised, and passing the “fit and proper” test is not enough to escape a probe, they add.

“We believe they have made a serious legal error in their duties. Indeed, we believe this is a sufficiently serious error to be open to judicial review should you not decide to make a reference on grounds of broadcasting standards,” the MPs tell Bradley in the letter.

“We urge you to recognise that the finding of significant corporate failure at Fox News, coming on top of the events at the News of the World, clearly meet the bar required [for an investigation by the CMA].

“Two such egregious examples of wrongdoing, and its persistence and toleration, suggests a failure of corporate governance which may well risk compliance with broadcasting standards and easily justify a CMA investigation.” The MPs said they were concerned about Sky News “being taken in the direction of Fox News”, a reference to the US channel’s rightwing editorial stance.

The Murdochs have offered to allay concerns about the future of Sky News by committing to funding it for at least five years and installing an editorial board with a majority of independent directors. However, Bradley concluded that this proposal was not enough for the deal to be approved without a CMA investigation.

Ofcom defended its report. A spokesperson for the regulator said: “We carried out detailed, evidence-based assessments, examining both broadcasting standards and media plurality aspects of the proposed takeover. The secretary of state will now decide on the next steps.”

Fox said in statement: “The criticism of the independent regulator and repeated efforts to politicise the quasi-judicial process are deeply disappointing and surprising, particularly in light of Sir Vince’s public comments following the recent UK election expressing ‘complete confidence’ in Ofcom’s review of our transaction.

“21CF remains committed to continuing to engage constructively with the regulatory process. Our investment in Sky underscores our profound confidence in the future of the UK as a global hub for creativity, and in light of the transaction’s benefits to the UK creative economy we would welcome a timely decision.”

The letter was sent by the MPs as Ofcom revealed that Sharon White, its chief executive, received a 15% salary increase last year. Ofcom’s annual report shows that White’s salary increased from £275,000 to £316,250 in the last financial year. Including bonuses, she collected £430,309, up from £375,200. Overall the regulator paid its senior management team £2.5m in 2016/17, up from £2.3m in 2015/16.

Ofcom became the independent regulator of the BBC this year.