The Whites are expected to replace current head coach Neil Redfearn and announce former Brentford and Wigan Athletic boss Rösler as their new head coach this afternoon at 3pm.
However, the 46-year-old was not their primary target, according to The Sun.
The newspaper claims that Warburton was the first to turn the job down after holding talks with the Brentford boss reluctant to work with Redfearn's coaching staff, while Poyet also declined the opportunity to take charge.
Warburton, who will leave Brentford this summer, has been linked with a host of managerial roles while Poyet has been out of work since being sacked by Sunderland in March.