Officers responded to a report of a dead body at the Hilltop Manor low-income senior apartment complex at 790 Ironwood Drive at about 11:35 a.m., Sgt. Jason Dwyer said.
When officers arrived, they found the stabbing victim, Dwyer said.
The man was later identified as 69-year-old Stanley Jacobson, who went missing in early February, according to police.
Police said last week that Jacobson, who suffered from dementia, had last been seen by a neighbor at his home on Ironwood Drive on Feb. 5.
His family reported him missing Feb. 18 and told investigators that Jacobson did not know how to read or write and had left his wallet and identification at home.
It was Jacobson's family members who discovered his body Thursday, Dwyer said.
Police said they have made an arrest in the case, but have not yet released the suspect's name.
At the time they announced Jacobson's disappearance, police said it was possible that he was in the company of a 46-year-old woman named Regina Butler, and that he may be driving a tan 1996 Ford Taurus with license plate No. 3PQF984.
Jacobson's death marked San Jose's eighth homicide of the year.
No one at Hilltop Manor was immediately available for comment.