The company reported a 7pc rise in revenues in the first half of the year to £192m as agents and developers spent more on listings. But it warned of a decline in customer numbers as high mortgage rates continued to drive a downturn in the property market.

The lull makes Rightmove an attractive target, given property market activity is forecast to pick up as interest rates continue falling.

Rightmove also has a familiar face in its leadership team: finance chief Alison Dolan previously held senior roles at Mr Murdoch’s News UK and Sky.

REA has until the end of September to make a formal offer.

The company warned there was no certainty of any deal and said shareholders did not need to take any action. While New Corp owns the majority of shares, its publicly traded stock fell as much as 8pc on news of the takeover interest.

The potential swoop underscores efforts by the Murdoch empire to expand away from its traditional media businesses, which include the Times and Sun newspapers and Fox News, into new areas as 93-year-old Rupert hands over control to son Lachlan.

Such efforts have been painful to date, with struggles to both realise profits from past investments and to convince investors of the value of its strategy.

News Corp last year abandoned talks to sell its US real estate business Move, Inc to rival CoStar in a deal worth a reported $3bn.

Activist investor Starboard Value subsequently urged the Murdochs to spin off REA. Starboard, which is run by US hedge fund manager Jeff Smith, said a sale of News Corp’s 61pc stake would unlock $7bn in value for shareholders.

The family was also forced to abandon plans to merge News Corp with sister company Fox amid complaints from shareholders that the combined group would be undervalued.

That came after a rare defeat for Mr Murdoch when he lost out to German media giant Bertelsmann in a $2.2bn bidding war for publisher Simon & Schuster.

A possible bid for Rightmove comes as the Murdochs are poised for a Succession-style legal battle for future control. Nonagenarian tycoon Mr Murdoch is seeking to change the family trust to hand sole control to Lachlan.

Such a move would shift power away from his other children – James, Elisabeth and Prudence – who are considered to have more liberal politics than their father. The legal dispute will go to trial in Nevada later this month.

A swoop for Rightmove would mark the latest takeover of a UK property company after rival OnTheMarket was snapped up by CoStar for £99m last year.

While Rightmove holds a market share of 86pc, CoStar’s takeover of OnTheMarket is expected to create a powerful rival to challenge its dominance.

Both companies also face competition from Zoopla, which was founded by British entrepreneur Alex Chesterman. Daily Mail owner Lord Rothermere was the largest shareholder in Zoopla until 2018, when the company was sold to US private equity firm Silver Lake Partners.

Any takeover bid would represent a major test for Johan Svanstrom, who took over as chief executive of Rightmove last year.

The company boasts a number of high-profile board members, including chairman Andrew Fisher, who was formerly chief executive of music app Shazam, as well as tech executive Jacqueline de Rojas and veteran media banker Lorna Tilbian.

Jessica Pok, an analyst at Peel Hunt, said: “It does not come as a surprise to us today that Rightmove has become an acquisition target, given the rating has been subdued for some time due to the negative sentiment on the UK housing market and concerns over competitive threats from CoStar/OnTheMarket.

“However, our belief, reflected by the takeover interest, is that the shares look attractive, given the stability of its core classifieds business and the growth opportunities in other revenue streams such as mortgages, commercial real estate and rental under the new chief executive.

“On top of that, with declining rates, we believe the UK property market has scope for improvement as we move into 2025.”

Rightmove was contacted for comment.