Williams, who is suspended indefinitely for his admitted role in overseeing a bounty system that offered Saints defenders cash for big hits, did not immediately respond to a phone message and email left with his foundation in Missouri on Thursday.
Pamphilon made the recording during a meeting before the Saints lost to the 49ers in a divisional playoff game in January.
When the New York Giants defeated the 49ers a week later in the NFC title game, several Giants players made similar comments about wanting to get hits on Williams, who fumbled twice in the game, because they knew he had previous concussions.
In Pamphilon's recording, Williams also tells his players to set their sights on running back Frank Gore, receiver Michael Crabtree and quarterback Alex Smith.
"We need to decide on how many times we can beat Frank Gore's head," he says.
"We need to decide whether Crabtree wants to be a fake (expletive) primadonna, or he wants to be a tough guy. We need to find out. He becomes human when we (expletive) take out that outside ACL," he said.
Williams made a hand signal suggesting he would personally pay for a ferocious shot on Smith, Pamphilon said.
"We hit (expletive) Smith right there," Williams said, pointing to his chin.
Pamphilon said Williams then rubbed his fingers together as one might do when doling out cash, saying, "I got the first one," which Pamphilon understood to mean the defensive coordinator had placed a cash bounty on Smith. "Go lay that (expletive) out."
The NFL has said Williams' bounty system offered off-the-books cash payments of $1,000 or more for hits that either knocked targeted opponents out of games or left them needing help off the field, and the Saints have been punished heavily for allowing such a program to endure for three seasons.
NFL Commissioner Roger Goodell also suspended Saints head coach Sean Payton for the entire 2012 season, while handing down additional suspensions of eight games to general manager Mickey Loomis and six games to assistant head coach Joe Vitt, who also coaches linebackers.
The Saints, meanwhile, were fined $500,000 and docked second-round draft picks this year and next.
The Williams recording was released on the same day that the Payton, Loomis and Vitt were in New York for an appeal hearing regarding their unprecedented punishments.
Pamphilon said Payton and Loomis were not in the room when the recording of Williams was made.
Williams can be heard using metaphors he has often used throughout his coaching career, such as, "kill the head and the body will die." That was Williams' way of urging players to disrupt opposing teams' star players with intimidating and nasty physical play. Another of Williams' mantras was that "respect comes from fear," which he repeats in the recording.
"We've got to do everything we can in the world to make sure we kill Frank Gore's head," Williams says. "We want him running sideways. We want his head sideways."