The Cavs, who have the ability to clear as much as $20 million in salary-cap space, are believed to be willing to start a deal at the maximum contract number of about $15 million per season.
The Cavs see Hayward as a complement to young franchise cornerstones Kyrie Irving and No. 1 overall draft pick Andrew Wiggins.
Cleveland, which got a commitment from Irving to sign a five-year, $90 million extension earlier this week, sees Wiggins as a shooting guard. Hayward would become the small forward.
The Cavs had been hoping to secure a meeting with LeBron James but haven't been successful and moved on to make Hayward their top free-agent target.
Hayward averaged 16.2 points, 5.1 rebounds and 5.2 assists for the Jazz last season in 77 games, all starts. He had seven games with double-digit rebounds and six with double-digit assists. He scored more than 20 points in 22 games.
Hayward was selected with the ninth overall selection in the 2010 draft out of Butler. For his career, he's averaged 12 points, 3.4 rebounds and 3.1 assists in 287 games, 179 of them starts.
No offer sheets can be signed until July 10.
ESPN.com's Marc Stein first reported the Cavaliers' interest in Hayward on Tuesday. Earlier Wednesday, Yahoo! Sports reported that the Cavs arranged to bring Hayward in for a visit.
The Associated Press contributed to this report.
Cavs Preparing Huge Offer For Hayward
Brian Windhorst discusses the Cavaliers' plan to offer a max contract to restricted free agent Gordon Hayward.