Media Down Under claimed that Cheika was offered the post with Australia's second-newest side late last week, although Force CEO Vern Reid refused to comment on the speculation.
Cheika, who has also been linked with a post at the Waratahs, has opted against heading to the west coast according to the Sydney Morning Herald.
The ex-Randwick star had been the heavy favourite to succeed Richard Graham since the latter left his post in April but the Force will now be under pressure to find another suitable candidate sooner rather than later.
With Cheika supposedly out of the running, 2005 Lions assistant Eddie O'Sullivan and former Bath boss Steve Meehan could re-enter the race but it is now unlikely an appointment will be made in the next few days as Reid had already urged patience even before this latest setback.
The news is a further blow to the Force's hopes of hanging on to star skipper David Pocock who has long been linked with a switch to other Australian sides.
Pocock is out of contract this summer and is being heavily tipped to join the Brumbies, although the Rebels have also declared a significant interest.
The Force are desperate to keep hold of the influential openside but the continued delay in appointing a new coach is unlikely to impress the man who captained the Wallabies against Scotland and Wales last month.