co-stars Benedict Cumberbatch and Martin Freeman are "frosty" towards one another, according to the Sun.
The newspaper quotes "pals" of the actors as saying that their friendship is "hardly close", raising questions over the future of the BBC One drama, which completed its fourth series on Sunday.
"Benedict and Martin aren't mates and they don't spend time together away from the show," according to the paper's anonymous source.
"They're professional and very polite to each other but there's not the warmth you'd expect after filming together for six years. There isn't a huge desire to come back for another season."
Questions continue to be raised over the future of the programme, given concerns over the film stars' continued availability; Cumberbatch also plays the superhero Doctor Strange in Marvel's film franchise, while Hobbit star Freeman is committed to several other projects, including forthcoming superhero film Black Panther.
A "well-placed BBC source" told Radio Times that there would not be another series of Sherlock next January, while adding that the broadcaster is "reasonably hopeful" the programme will return for 2018.
However, Sherlock writer and co-creator Steven Moffat has suggested that Sherlock may have come to an end. "I don't know how long we can keep it going. I'm personally willing but I'm hardly the main draw," he said in July. "I would be moderately surprised if this was the last time we ever made this show. But it absolutely could be."
The final episode of the series aired last weekend in the UK, where it faced low ratings following a leaked version of the episode coming online.
The fourth series starts on TVNZ 1 January 29.