Stewart would not reveal specifics of the Diamondbacks' offer, but MLB.com reported Sunday that Cueto rejected a six-year, $120 million deal.
Teams in the market for Cueto believe that he is seeking a deal worth between $140 million and $160 million, a source told ESPN's Jerry Crasnick.
Stewart told the Arizona Republic he met with Cueto in the Dominican Republic and made the right-hander an offer. The general manager was joined by chief baseball officer Tony La Russa and club president Derrick Hall for the Diamondbacks' pitch, a source said. Stewart also said he met with Cueto's agent, Bryce Dixon, at Chase Field.
Cueto is one of multiple high-profile starting pitchers, along with David Price and Zack Greinke, available in free agency this offseason. The Detroit Tigers made the first big splash earlier Sunday by agreeing with right-hander Jordan Zimmermann on a five-year deal worth $110 million, a source confirmed to ESPN.
"When you've still got the two big guys who haven't done anything yet, Price and Greinke, you kind of want to wait," Stewart told the Republic. "Depending on how [Dixon] sees his client, and he probably see his client above Zimmermann, it puts you in a situation where you want to wait and see what happens."
Stewart told the paper the Diamondbacks still would like to sign Cueto, but he acknowledged that team ownership would have to approve a more lucrative offer.
"Having a conversation with [Cueto] gave us an idea of what's going to happen and what he's thinking," Stewart said. "We're going to have to wait and see how this thing plays out."
Cueto, who turns 30 in February, helped the Royals win the World Series after being traded to Kansas City by the Reds earlier this year. In 2015, the All-Star went 11-13 with a 3.44 ERA in 32 starts with the two teams.