The search was carried out in “very challenging” conditions due to the river flow and level being heightened after the heavy rain over the weekend, police said.
Mr Voyce, from Truro in Cornwall, played for Wasps, Bath and Gloucester before retiring from the sport in 2013.
He previously worked for the bank Investec and has been living in Alnwick since 2020.
Writing on X on Wednesday, Rugby World Cup winner Lawrence Dallaglio said: “It’s the most horrific news and all I would say at this juncture is that please let us all respect Tom’s family and their suffering right now and keep them in all our thoughts.”
Matt Dawson, who played alongside Mr Voyce at Wasps between 2004 and 2006, previously said he could not “express his sadness”.
Premiership Rugby said it was “devastated” and thinking about Mr Voyce’s loved ones.