Before he was elected to Congress, Rogers was an FBI Special Agent from 1989 to 1994.
"Chairman Rogers exemplifies the principles that should be possessed by the next FBI Director," FBIAA President Konrad Motyka said in a statement. "His unique and diverse experience as a veteran, FBI Agent and member of Congress will allow him to effectively lead the men and women of the Bureau as they continue their work to protect our country from criminal and terrorist threats."
Motyka added that Rogers "has never forgotten the men and women of the Bureau and has been a champion for Agents since his election to Congress."
In response, Rogers said he was "honored" by the FBIAA's endorsement.
"I am honored to have the confidence of the men and women of the FBI's Special Agent community, and am humbled by their endorsement of me to lead the Bureau," Rogers said in a statement. "The next generation of FBI leadership must recognize how essential Special Agents are to the Bureau's core mission. In whatever capacity I serve the public, my focus will always be to ensure we are in the best position possible to keep America safe."
Rogers has also been considering a run for the Senate, and would likely be Republicans' strongest candidate. If he declines a run to take the FBI job the GOP's chances at defeating Rep. Gary Peters (D-Mich.) in the open seat.
The Obama administration reportedly wants to find a successor to Mueller before Congress goes on summer recess so the nominee can be thoroughly vetted and confirmed. Other names reportedly under consideration include Lisa Monaco, who advises President Obama on counterterrorism.
Updated at 6 p.m.
Cameron Joseph contributed