Jurich told ESPN that Petrino told him, "I don't need anything in writing; I can trust Tom."
Petrino's contract is expected to be a seven-year deal, a source told ESPN's Joe Schad.
Louisville has scheduled a 10:30 a.m. ET news conference Thursday to formally introduce Petrino, who will be serving his second stint as the Cards' coach. The news conference will follow a special meeting of the school's board members, who must approve the hiring.
It's unknown when Petrino's contract with Louisville will be finalized, Jurich said.
Among the other candidates was current Louisville defensive coordinator Vance Bedford, who congratulated Petrino on his Twitter account, saying, "I want to thanks the card nation for your support. Coach petrino will do a great job. Stand behind the team. Go Cards."
Petrino, 52, got his start as a head coach at Louisville, leading the Cards from 2003 to 2006 before leaving for a brief stint as coach of the Atlanta Falcons.
After leaving the Falcons, Petrino returned to the college ranks with Arkansas in 2008 but was fired in April 2012 for a "pattern of misleading behavior" following an accident in which the coach was injured while riding a motorcycle with his mistress, a former Razorbacks volleyball player.
In the wake of the scandal, Petrino signed a four-year contract with Western Kentucky in December 2012. The deal has a base salary of $850,000 and includes clauses in which the oft-traveled coach must repay the school $1.2 million in six installments beginning a month after any early exit, while the program that hires him must agree to schedule a two-year, home-and-home series with WKU.
Petrino was 41-9 at Louisville, leading the Cardinals to a 12-1 mark and their first BCS bowl victory in the 2006 Orange Bowl.
He is 83-30 overall as a college coach, including an 8-4 this season with Western Kentucky.
Strong, who guided the Cardinals to a 12-1 record this season, was introduced as Texas' coach Monday.
The Associated Press contributed to this report.