Barraclough held a 15-13 record as a Nittany Lion.
Lee will take over for Jimmy Kennedy, who left Penn State to work for David Taylor at Oklahoma State.
Kerkvliet is set to return for his sixth and final year of NCAA eligibility.
"I was offered a big pay day too," Starocci wrote on Twitter. "I said go wipe your ass with that money. When it comes to money people switch like the weather. Not me."
Taylor wrestled for Penn State from 2009-14 and amassed a 134-3 record.
"With one year of eligibility left, I want to end my collegiate career on my terms. Healthy, in style, and in a dominant fashion."