Ten years ago Keir Starmer attempted to send Rebekah Brooks to prison for phone hacking.

Now Starmer could cause another headache for the boss of Rupert Murdoch’s British media empire. She has to work out how her Tory-backing newspapers – which include the Sun and the Times – handle the growing popularity of the man who is favourite to become the next prime minister.

One question being asked by journalists at News UK’s newspapers is whether the history between Brooks and Starmer will have any impact – and whether those tensions can be squared with the growing desire for a Labour government among the consumers of its outlets.

“News UK operates a bit like the Chinese Communist party – with a long memory and a very long plan,” said Tom Baldwin, a former Times political journalist who served as Labour’s director of communications under Ed Miliband. “The Times has got a problem. A plurality of its readers would want Keir Starmer to be prime minister at the moment, but it doesn’t have a columnist who is speaking to the Labour party.”

Brooks was ultimately found not guilty of all charges at her Old Bailey trial and returned to her old job as chief executive of News UK. But Starmer – then the director of public prosecutions, overseeing criminal prosecutions in England and Wales – stood by his belief that it had been right to prosecute Brooks: “She put her case and she answered it and we must respect that fully. But the deeper question is, is anybody above the law? This has answered that.”

There is also longstanding bitterness among some older journalists at the Sun towards Starmer, after he restarted the Operation Elveden prosecutions over payments to public officials for stories. Of the 29 cases against journalists – many at News UK outlets – there was only one where the conviction stood.

Despite this history, there are signs that a subtle rapprochement may be on the cards, exacerbated by dismal Conservative poll ratings. Although there is no sign a formal change is coming, one Times journalist suggested tectonic plates were shifting ahead of the next general election: “No decision has been made but minds are opening.”

This is backed by internal market research, briefly posted on a staff intranet before being deleted, showing growing support for Starmer among the Times’s well-off readership. The outlet’s audience has been substantially pro-remain since the EU referendum, loathing Boris Johnson and Liz Truss, but does have more support for Rishi Sunak. Working out what to do with the Sun, where readers are more solidly supportive of Brexit, is more challenging.

Baldwin said the impact of Murdoch’s media endorsements had probably always been overstated, especially with the waning influence of newspapers, but few doubt the decision ultimately comes from the top: “At the Times there is usually a kind of veil cast over it, so it doesn’t look like it’s proprietorial interest. Great play is made of having a meeting with the political writers about what to do – although everyone assumes the decision has been made.”

Murdoch’s newspapers have often been flexible when it comes to political endorsements and reflecting the views of their audiences. News UK likes to back political winners and supported Labour at general elections between 1997 and 2005, after Blair made a dash to Australia to win the approval of Murdoch.

While Starmer said during his leadership campaign that he wouldn’t talk to the Sun, he has since written an editorial for the outlet. There is pressure from within Labour for more interaction with the newspaper, partly to reach its readers – and partly to show a symbolic break with the Jeremy Corbyn era. This is despite the Sun facing new allegations that phone hacking took place while Brooks was editor. News Group Newspapers has always maintained that phone hacking did not take place at the Sun.

News UK declined to comment on the political direction of its papers. However, shortly after the Guardian made inquiries, the Times ran an editorial assessing Starmer’s leadership. It praised his overall direction but said Labour was surging mainly due to the “government’s mistakes rather than its own qualities” – and criticised his positions on transgender issues, press regulation and rail strikes.

Yet there is growing confidence in Labour circles that the party will not face the same level of hostile media coverage that Corbyn endured. As one senior party official, reflecting on the Sun’s record of endorsing the victorious side in every election since 1979, put it: “Is the Sun not going to back a winner for the first time ever?”