The 22-year-old winger has struggled to hold down a first-team place at Bramall Lane, and was loaned out to York City in 2012.
He also completed a loan stint with Norwegian side HamKam earlier this year, before being recalled by the Blades.
"We are very pleased to have concluded the transfer and feel it is a good solution for us," club director Tom Rune Espedal told Stavanger Aftenblad.
"Players get a good education in England, but it is also positive that he has got experience from the first division in Norway and is ready to fight for a place in our side."
Tonne has signed a two-year contract with Sandnes Ulf.