The victim was in the dining hall requesting to go back to his cell, when officers saw him bleeding.
The inmate's wounds were "impressive" and he was airlifted to a hospital to threat them, the prison's spokesperson, Darren Chamberlain, said.
On Wednesday, the 28-year-old inmate was reported to be recovering and in a stable condition. His name will not be released because officials fear for his safety.
Chamberlain said the inmate was stabbed with one piece of metal about 3 p.m. Tuesday. The weapon was recovered at the dining hall shortly after the stabbing.
The attempted homicide is being investigated and three suspects have been detained in the Administrative Segregation Unit.
Although there are "too many unanswered questions" on the attack, officials are not ruling out a gang-related motive.
According to Chamberlain, the victim is associated with the Bloods criminal street gang founded in Los Angeles, where he committed the crimes he's serving time for.
The inmate is serving a 10-year sentence for burglary and assault with force, Chamberlain said. His earliest release would be in October 2015.
He began his sentence in March 2007 at the California Department of Corrections and Rehabilitation and was transferred to SVSP on August 2010.