The Rangers exercised their $9 million option, but Nathan had the right to decline it because he finished more than 55 games in 2013.
Texas does not owe Nathan a $750,000 buyout.
Nathan told ESPNDallas.com on Thursday that he was looking for at least a two-year deal.
"There are other teams that have a bigger need in the bullpen and put more resources than we are," Rangers general manager Jon Daniels said. "Joe's been outstanding for us on and off the field. He is as advertised, but we have some other areas that are a higher priority for us to address."
Nathan had 43 saves and a 1.39 ERA in 67 appearances for the Rangers in 2013. It was the second-highest save total in Rangers history. He made the All-Star team the past two seasons in Texas.
"The bullpen in general is a strength for us and an area of depth," Daniels said. "I would expect our closer ultimately to be somebody already in the organization. I don't know that we're going to necessarily name one. We have a number of guys internally that have done that for us before."
That includes Joakim Soria, a former All-Star closer, and Neftali Feliz, who closed for the Rangers for two seasons before moving into the rotation. Tanner Scheppers is also a candidate after performing well in the eighth-inning setup role last season.
Feliz is continuing his rehab and preparing to be in the club's bullpen next year.