Clarkson agreed to a four-year, $52 million deal to stay in Utah, his agent, Rich Paul of Klutch Sports, told ESPN's Adrian Wojnarowski.
The Jazz also agreed with Favors on a three-year deal worth $30 million, his agent, Wallace Prather, told ESPN's Bobby Marks. The deal includes a player option for the final year, Prather said.
Clarkson, 28, stepped up for the Jazz in the seeding games and playoffs after Bojan Bogdanovic was ruled out ahead of the NBA's resumption of play last season.
The shooting guard averaged 16.7 points per game off the bench and shot 46.4% from the field in the Jazz's first-round loss to the Denver Nuggets.
A dynamic bench scorer, Clarkson makes a healthy number of 3-pointers, can attack closeouts and has flashed the ability to make plays for others. However, he has proved to be a liability on defense, which limits his ceiling to that of a role player.
Clarkson, a second-round pick in the 2014 NBA draft, has spent time with the Lakers, Cavaliers and Jazz. He holds career averages of 14.8 points, 3.1 rebounds and 2.6 assists per game.
Favors, 29, spent nine seasons in Utah before signing a one-year, $17.65 million deal with the New Orleans Pelicans last season.
Despite the team's bevy of young talent, Favors started nearly all the games he played and provided the Pelicans with a steady hand at center. New Orleans was 5.4 points better defensively when Favors was on the court and 2.8 points better offensively per 100 possessions.