Three arbitration-eligible players -- closer Aroldis Chapman, setup man J.J. Hoover and shortstop Zack Cozart -- were tendered offers, as expected.
Mattheus, 32, was claimed off waivers from the Angels on May 13 and posted a 4.09 ERA in 55 innings over 57 appearances. He retired 39 of his 57 first batters faced and stranded 13 of his 19 inherited runners.
Villarreal, who turns 28 on Dec. 9, split the 2015 season between Cincinnati and Triple-A Louisville. In 29 big league games, often as a long or middle reliever, he had a 3.