Houston police announced Wednesday that Pappas is their prime suspect in Hausknecht's murder.
Joseph James Pappas, 62, is accused of following Dr. Mark Hausknecht on his bicycle on July 20 and fatally shooting him three times as he cycled to work in Houston, Texas.
Houston Police Chief Art Acevedo said that authorities have an arrest warrant for Pappas, who is armed, unsafe and potentially suicidal, NBC reported.
Police have identified a suspect in the shooting death of a cardiologist who once treated George HW Bush as the vengeful son of a woman who died on the doctor's operating table 21 years ago.
Houston Police said they searched the home of Joseph James Pappas, the suspected gunman and are actively looking for him.
The investigators obtained a search warrant and entered Mr Pappas' home at 4am on Wednesday, but no-one was present.
Anyone with information should call Crime Stoppers of Houston at 713-222-TIPS (8477) or the Houston Police Department homicide division directly at 713-308-3600. Police believe that he might be suicidal, Acevedo said.
"The suspect had not been heard from by people that know him for over 24 hours and there is a text that he sent someone that said he was going to kill himself", the police chief said.
"I never expected this at all", she said.
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"We found plenty of evidence that shows an extreme interest by (Pappas) in the doctor", Acevedo told CNN on Thursday.
Acevedo says they believe the motive may be that Pappas' mother died while undergoing surgery 20 years ago under the care of Hausknecht. She described Pappas as smart and helpful. "There was a lot of planning and sadly some skill".
No disciplinary action had ever been taken against Hausknecht, according to the Texas Medical Board.
Pappas had worked from 1983 through 1995 as a peace officer with two Houston-area constable offices.
News of his death made national headlines and prompted heartfelt responses from the medical community, other local hospitals and former U.S. President George H.W. Bush, the doctor's former patient.
Friends and colleagues remembered the doctor as a quiet, generous and kind person.
That phone number is linked to recent listings on a firearms auction website for several guns, ammunition, tactical vests and ballistic plates for auto doors.
Meantime, the community was mourning the prominent doctor whose life was cut short after nearly four decades in practice.