The York St John University student disappeared during a night out with friends in the city in January.
It was feared she became separated from friends and fell into the River Ouse near York's Lendal Bridge.
The woman's body was spotted at 3.30pm yesterday by a member of the public. However, while Megan's family have been told of the discovery, the body has yet to be formally identified.
A spokesman for North Yorkshire Police said: "The body of a woman has been found in the River Ouse at Acaster Malbis near to the Ship Inn.
"Police received a call at 3.27pm on Sunday 2 March 2014 from a member of the public who was walking in the area.
"Formal identification has not yet been completed, therefore we cannot confirm the identity of the deceased at this stage. The family of Megan Roberts have been informed of the discovery."
Megan vanished on 23rd January 2013 after enjoying a night out with her friends. However it was not realised she was missing for several days, as her housemates believed she had returned home to see her family.
Megan was last seen on CCTV on the corner of Rougier Street and Lendal Bridge in York at around 2.20am, leading police to surmise that she had become separated from her friends while under the influence of alcohol and fallen into the river.
Megan's mother, Jackie Roberts, also said she believed the disappearance of her "beautiful daughter" appeared to be the result of "a tragic run of events with no one to blame".