The Reds announced Bray's deal for $645,000 on Tuesday.
Cueto is asking for $3.9 million and the club offered $3 million. The right-hander made $445,000 last season, when he went 12-7 with a 3.64 ERA.
Volquez is asking for $2 million and the Reds proposed $1.3 million. He returned from reconstructive elbow surgery and made a dozen starts, going 4-3 with a 4.31 ERA.
The Reds avoided arbitration with NL MVP Joey Votto, who signed a three-year, $38 million deal on Monday.