The D-backs acquired the right-handed reliever from the Cubs during the Major League phase of the Rule 5 Draft in December. Mateo, 29, went 2-0 with one save and a 1.74 ERA in 24 games with Triple-A Iowa, Double-A Tennessee and Rookie Arizona League Cubs last season.
He went 3-1 with a 1.71 ERA with 24 strikeouts in 21 innings for the Estrellas Orientales in the Dominican Winter League and later pitched for the Aguilas Cibaenas during the playoffs. Mateo missed all of the 2012 season and part of '13 because of Tommy John surgery.
"If I don't make this team, they have to send me back to Chicago because of Rule 5, but I'm not going back to Chicago," Mateo said. "I have to take advantage of the opportunity here."
The D-backs could offer Mateo back to Chicago or negotiate a trade with the Cubs to keep him in the organization if he does not make the team. The D-backs already feature several veteran right-handed bullpen options.
Mateo went 1-3 with a 5.04 ERA in 44 2/3 innings in 44 relief appearances for the Cubs in parts of the 2010 and '11 seasons.
Jesse Sanchez is a national reporter for MLB.com. Follow him on Twitter @JesseSanchezMLB. This story was not subject to the approval of Major League Baseball or its clubs.