TVWise claims that the two shows will not be returning for further runs.
Based on the Lynda La Plante novels, Above Suspicion starred Ciaran Hinds as Detective Chief Superintendent James Langton and Kelly Reilly as Detective Inspector Anna Travis.
The latest series, subtitled 'Silent Scream', focused on the murder of a film star. The fourth and apparently final run finished on January 23.
The drama series Kidnap and Ransom starred Trevor Eve as an international hostage negotiator. It ran for a total of six episodes and two series on the channel.
Eve was previously quoted slamming the BBC for focusing its energy on shows like Doctor Who.
He said: "All they are interested in are spinoffs from Doctor Who and [shows similar to] Doctor Who, like Sherlock... It's fine as far as it goes, but Doctor Who is a 40-year-old children's programme."
The cast and creators of both shows have yet to speak out on the reports.