West Mercia Police launched an investigation after the discovery at Cressage last month.
A member of the public alerted the police after spotting what they believed were human remains in the river between Shrewsbury and Much Wenlock on Saturday, May 21.
Officers said the death was being treated as "unexplained", with all lines of enquiry being looked at.
After attending the scene, police said that it was believed the person had been in the water for a significant amount of time.
In an update the force confirmed that enquiries are ongoing and "may take some time".