Desmond's Northern & Shell group, owner of the Express and Star papers, is understood to be targeting savings of anywhere between £8m and £18m at the titles.
The Guardian claims that executives at the firm are also reviewing whether the Sunday edition of the Daily Star is now viable following the return of News International to the market with the Sun on Sunday last month.
A meeting is due to be held this afternoon between Northern & Shell executives and the trade unions, after which the scale of cuts is expected to become clear.
The Daily Star Sunday was one of the biggest beneficiaries of the closure of the News of the World in July 2011, and is still selling at around 170,000 copies more a week than it was before the rival paper fell.
However, the tabloid is thought to be under new financial pressure after dropping its cover price to just 50p to compete with Rupert Murdoch's Sun on Sunday.
Speculation last year was that Desmond - also the owner of Channel 5 - was reviewing the potential sale of his celebrity magazines such as OK!, but a source said that his attention now appears to be back on the papers, and he is demanding "action".
"The company has made it clear nothing is safe, everything is vulnerable and comments about closing titles have been made," the source told The Guardian.
"Regardless of where the cuts level comes out, it is hard to see how they will go through without substantial changes to the business."
Northern & Shell refused to comment on the level of cuts ahead of today's meeting.
A spokeswoman said: "There is a meeting between management and unions and we have nothing to say before then, because it would be discourteous to the unions and the process we have been involved in."
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