The force said officers received a report on Sunday morning that Voyce, who played for clubs including Bath and Wasps, had not returned home after an evening with friends. Northumbria police said it is believed he attempted to cross Abberwick Ford in his car, which was then pulled along with the current of the river.
The car has since been recovered but officers did not find Voyce and it is believed he was swept away while attempting to escape and has died.
Ch Supt 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 inquiries have been ongoing since concerns were raised for Tom, including deploying specialist teams to search for him."
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