The team made the announcement Thursday, a day following the end of the season in which the Suns went 19-63.
"[Jones] is instilling the same championship culture and standard that he experienced on multiple occasions as a player," Suns owner Robert Sarver said in a statement.
"I have the utmost confidence in James as the leader of our basketball operations moving forward, and we are aligned in the ultimate goal of one day bringing an NBA championship to Phoenix," Sarver said.
Jones had filled the role of Suns general manager on an interim basis for a majority of the past season.
Sarver said Jones will oversee all basketball operations, and that Bower and assistant general manager Trevor Bukstein will report to him.
Jones took over the general manager role in October when the Suns fired Ryan McDonough eight days before the start of the season.