With the Bruins winning their first College World Series title last month, UCLA has offered to extend the coach's contract through 2025, pending approval by the university's board of regents, according to a person with knowledge of the negotiations.
Savage did not immediately return a phone message seeking comment, but the person said USC had come after the coach "very hard" and with a "spectacular offer."
Savage was paid more than $300,000 annually, including bonuses, by the terms of his previous UCLA contract, which ran through 2017. USC's offer was for more than $1 million per year, which would have made Savage the highest-paid coach in the nation.
Savage, 48, has been UCLA's coach since 2005, but he is also a former USC pitching coach.
The Bruins have participated in the NCAA playoffs in seven of the last eight years and advanced to the College World Series in three of the last four.
UCLA finished third in the Pac-12 Conference this spring, but won 10 consecutive playoff games. In Omaha, the Bruins gave up four runs in five games with an impressive display of pitching and defense.
Without a slugger or .300 hitter in the lineup, UCLA also generated enough offense with timely hitting and by executing bunts, sacrifices and hit-and-run plays.