Ministers are facing pressure to appoint a former Conservative cabinet minister as the new chair of the media regulator Ofcom, as he battles for the job against a Labour peer.

The job of running the regulator has become a key post in public life, as concern over online content has grown rapidly, alongside the rise of more politically partisan broadcasting. No successor has yet been named to replace Michael Grade, the former BBC chair who has just weeks left in the job.

It is understood there is a concerted push to convince Liz Kendall, the technology secretary, to hand the job to the former culture secretary and sitting Conservative MP Jeremy Wright, who was involved in the drafting of new laws tackling harmful online content.

The legal pitfalls Ofcom is facing in implementing the new Online Safety Act have led to claims of paralysis at the regulator.

Wright has been shortlisted alongside the businessman and former Channel 4 chair Ian Cheshire and the former MP and Labour grandee Margaret Hodge. Insiders believe the race for the job is now between Wright and Hodge.

Hodge has been seen as the favourite to be appointed by the Labour administration. As chair of the public accounts committee, Westminster’s spending watchdog, she built a fearsome reputation for attacking big tech over its tax bill. She has also previously suggested banning online anonymity and making social media directors personally liable for defamatory posts.

However, several sources involved in online safety said Wright should be given the job. They pointed to his legal background as a KC and his knowledge of the details of the Online Safety Act. They also believe he will take risks in confronting big digital platforms.

“Jeremy is the person who can really achieve the strategic reset that is required,” said a source familiar with the process. “He understands the workings of the act, which makes him perfectly well placed.”

Another said: “Ofcom should be testing its powers effectively and being more ambitious … Jeremy would be seen by civil society generally … as being the preferred candidate.”

Appointing a Conservative would prove controversial within Labour. However, some believe Wright would have the political freedom to take on the hostility to the new online safety laws from Reform UK and some Conservatives.

The former foreign secretary William Hague said Wright would be “a very clever appointment”. “Ofcom will need cross-party support to do its work well, so appointing an opposition member would be wise,” he said. “And he also has credibility on the issues from his ministerial experience.”

Whatever the decision, the delay in appointing a new chair is causing concern. “This is a crucial appointment that will signal whether the government is serious about achieving a strategic reset of the UK’s online safety agenda,” said Andy Burrows, the chief executive of Molly Rose Foundation.

“The new chair must address fundamental flaws in Ofcom’s implementation of the Online Safety Act and prove capable of restoring the frayed support and confidence of civil society. Ultimately the successful candidate must be able to successfully steer Ofcom towards implementing and enforcing regulation in the way that parliament envisaged when the act was passed.”

Imran Ahmed, the chief executive of the Center for Countering Digital Hate, said: “A delay in selecting a new chair for Ofcom would leave Britain at risk, with a dangerous leadership vacuum at Ofcom. Social media is getting worse, not better. Ofcom is already struggling to deliver on its online safety mandate and needs swift and decisive direction from a knowledgeable, experienced, and digital-savvy chair committed to protecting Britons from online harms.”

A government source said a decision would be made “very soon”.

An Ofcom spokesperson said: “Since the Online Safety Act began to take effect last year, we’ve already driven meaningful change, like age-gating pornography, requiring platforms to tackle illegal content and harms to children, and opening investigations into nearly 100 sites where we have concerns.

“But there is more work to do, and we look forward to working with whoever government appoints as our next chair to make life safer online.”