The publisher's sales arm, The Publishing Consultancy, has promised agencies that Cut will deliver a settle-down circulation of 100,000 or they will receive a refund.
Unofficial retail industry sales figures for Cut's first full-price issue (its first two sold at half price, 50p) suggest that it has achieved a sale of just 30,000.
The unofficial figures also show its first two discounted issues sold just 7 per cent of their print run, with 50,000 copies sold from a print run of 700,000.
However, Simon Young, the managing director of TPC, argued: "It's nigh-on impossible to give an accurate figure - it's too early."
Cut entered the men's weekly market last month, eight months after the launch of IPC Media's Nuts and Emap Consumer Media's Zoo. Nuts has a circulation of 290,337 and Zoo of 200,125.
Young said: "We are seeing agencies and viewing 100,000 as a ballpark figure but it is a softer selling process than Nuts and Zoo and the product is very different."
In a bid to reach its 100,000 target, Cut will use product sampling.
While Bauer is still committed to the title, industry sources argued it will be hard for Cut to recover from such a poor launch period.
The quality of Cut's paper stock and some of its content has been criticised, as has its decision not to run cover images of scantily clad women. Last week's issue featured the West Indian cricket captain, Brian Lara, on the cover. However, media buyers said there are signs of improvement in its look and feel, especially in the quality of paper and reproduction.
Bauer is attempting to build a circulation for Cut with a lower marketing budget than the £8 million that is being invested behind each of its men's weekly sector rivals, Nuts and Zoo.