According to a witness, Joshua Dean Squires lost his balance while trying to cross the rocks at Pillar Falls, and got swept into the river.
The other man tried to reach him, but he also slipped and was swept downstream by the current.
That man survived, but Squires is believed to have drowned, according to the Twin Falls County Sheriff's Office.
The 22-year-old victim is originally from Gooding, but was living in Twin Falls, sheriff's office informed.
Rescuers could not reach the spot where Squires' body is presumed to be due to swift currents.
The effort to find the body will start again Thursday afternoon.