Gruden joined the Jaguars' staff this off-season after he was fired in October by the Washington Redskins. Minshew, 24, posted a 6-6 record and threw 21 touchdowns and six interceptions in 14 games last season for the Jaguars.
He was thrown into the role after free agent signing Nick Foles sustained a broken collarbone in Week 1.
The Jaguars traded Foles to the Chicago Bears in March, a move that showed the franchise's confidence in Minshew as their starter in 2020.
"Well, it better mesh," Gruden told reporters, when asked about his relationship with Minshew. "It's my job to make it mesh, you know, to take on the personality of the quarterback and get him to understand what we're trying to get accomplished on the play-to-play, game-to-game, week-to-week basis.
"It's going to be critical."
Minshew joined the Jaguars as a sixth-round pick in the 2019 NFL Draft. His teammates have already praised him this off-season for his leadership ability.
Minshew has taken charge to help his teammates learn Gruden's new offensive playbook while Jaguars players meet virtually through video conference calls.
Gruden praised Minshew's "competitive spirit" and said he has seen it when watching game footage of the Jaguars quarterback.
"Now it's just a matter of him getting some general knowledge of our offense and me figuring out what he likes, what he doesn't like, what makes him tick and go from there," Gruden said.
"I'm pretty easy to get along with, I know he is too, so I think it will be a great relationship between the two of us."
The Jaguars also acquired backup quarterback Josh Dobbs last year in a trade with the Pittsburgh Steelers. Glennon and Dobbs will likely battle for the No. 2 job this off-season.
Gruden spent six seasons as the Washington Redskins' head coach. He worked with several quarterbacks in Washington -- Robert Griffin III, Kirk Cousins, Colt McCoy, Alex Smith, Mark Sanchez, Josh Johnson, Case Keenum and Dwayne Haskins.
Cousins was the only quarterback to go to a Pro Bowl while under Gruden, an honor he earned in 2016.
Gruden had a successful run as the Cincinnati Bengals' offensive coordinator before he joined the Redskins in 2014. Former Bengals quarterback Andy Dalton went to the Pro Bowl in 2011 under Gruden. The Bengals went 30-18 in three seasons with Gruden as offensive coordinator.
The Jaguars are scheduled to face the Carolina Panthers in their first pre-season game at 7:30 p.m. EDT Aug. 13. They are to begin the regular season against the Indianapolis Colts on Sept. 13 in Jacksonville, Fla.