Nathan was released along with fellow reliever Matt Albers.
"There's 29 other teams," Nathan told reporters, according to MASNSports.com. "I knew every time I took the mound this spring, it wasn't just to try to make this club but showcase for other teams. It was important for me to get out there, even yesterday, in a spot where they probably already knew I wasn't going to make this team.
"They still got me on the mound to pitch, and I appreciate that. [Manager Dusty Baker] helped me out even when he probably knew I wasn't going to be on this club, to send me out there and get another look with other clubs."
Nathan has had two Tommy John surgeries in his career, most recently in 2015, when he was with the Detroit Tigers.
His 377 career saves rank second among active pitchers and eighth in major league history.
The Associated Press contributed to this report.