Specialist officers from the marine unit found the body, which is yet to be identified, in the river near to the ford.
Voyce turned out nine times for England between 2001 and 2006, and retired in 2013 having played for Bath and London Wasps.
Over the weekend, the river level was “significantly heightened due to the exceptionally heavy rain”, said Northumbria Police Chief Superintendent Helena Barron.
Storm Darragh, which hit the UK over the weekend, left two confirmed dead when trees fell on their vehicles in northwest and central England.