Police said they received a report on Sunday that the 43-year-old had not returned home after an evening with friends.
A force spokesperson said: "At this time, it is suspected Tom has attempted to cross Abberwick Ford in his car, which has then been pulled along with the current of the river.
"Officers have since recovered his car, but sadly Tom is yet to be found.
"It is believed in his attempts to escape he has been swept away and tragically died."
Cornwall-born Voyce began his professional career at Bath but enjoyed his most successful years at Wasps, then based in London.
A prolific try scorer from wing or fullback, Voyce was part of the all-conquering Wasps team that did the Premiership and European Cup double in 2004, and Voyce helped them win both titles a second time but missed their third European final victory through injury.
He won nine England caps between 2001 and 2006 and retired from the game in 2013.
Chief superintendent Helena Barron, from Northumbria Police, said: "This is an extremely tragic incident, and our thoughts are very much with Tom's loved ones at this time.
"Our officers continue to support his family and we would ask that their privacy is respected.
"Extensive enquiries have been ongoing since concerns were raised for Tom, including deploying specialist teams to search for him."
England rugby expressed its sympathies to Voyce's loved ones.
"Our thoughts and prayers are with the family and friends of Tom Voyce at this extremely challenging time," England rugby said in a post on X.