Zunino, 21, was the third overall pick in last month's 2012 First-Year Player Draft. He agreed to a $4 million signing bonus, MLB.com has learned.
Zunino will be introduced to the media at a Tuesday news conference. He's been in Seattle with his family the past two days talking with team officials and undergoing a physical.
Zunino is the son of Reds scout Greg Zunino. He was named a first-team Louisville Slugger All-American by Collegiate Baseball and won the Dick Howser Award as the nation's top college player and the Johnny Bench Award as the top catcher.
He's headed to New York on Wednesday for the presentation of the Golden Spikes Award, where he is a finalist along with pitchers Mark Appel of Stanford and Carlos Rodon of North Carolina State.
The Mariners have now signed 29 of their top 30 picks, with the only unsigned player being eighth-rounder Nick Halamandaris, a high school first baseman from California.