Police said the search was carried out in “very difficult” conditions due to rising river levels and river flows following heavy rains over the weekend.
Voyce, from Truro, Cornwall, played for Wasps, Bath and Gloucester before retiring from the sport in 2013.
He previously worked at Investec bank and has been living in Alnwick since 2020.
Among those from the rugby world who paid tribute was former England international Mike Tindall, who played with Mr Voyce at Gloucester.
A post shared on his podcast The Good, The Bad and The Rugby reads: “He epitomized rugby as a game for all shapes and sizes, punching well above his weight.
“His tackle-breaking ability and durability were exceptional. Off the field he was a great character.”
His co-host, former England star James Haskell, also paid tribute, adding: “He was a teammate, a friend and an incredible player – someone I admired who always punched above his weight.
“He will be greatly missed by all.”
