A male suspect is in custody, but no charges have been filed, a department spokesman said.
Director Toney Armstrong identified the slain officer as Terence Olridge, 31. Olridge had been with the Memphis Police Department since September 22, 2014. He leaves behind a fiancee who is four months pregnant, Armstrong said.
Police responded to the shooting at 12:52 p.m. in a residential neighborhood, Armstrong said. Olridge was found and taken to Regional Medical Center, where he died.
"Everything is very, very sketchy," he said. "We're at the beginning phase of this investigation. There are more questions than we have answers right now."
Olridge's uncle said the officer was on his way to work in his uniform when he was shot while in his car, CNN affiliate WREG reported. Olridge was able to crawl to a house to get help, the uncle told the station.
The suspect's sister told CNN affiliate WMC her brother was washing his car and got into an argument with Olridge.
Neighbors said they heard at least 15 gunshots, WMC reported.
Mayor A.C. Wharton Jr. appeared at a news conference with Armstrong and said, "There's nothing I can add other than pray for the family of this young man, and I emphasize he was a young man, relatively new to the department, so pray for him, his family, and for the department and for the city as a whole."
Olridge is the fourth Memphis police officer to be killed in the past four years.
The last incident was August 1, when Officer Sean Bolton, 33, was shot and killed during a traffic stop. Armstrong said Bolton apparently interrupted a drug transaction.
A suspect in Bolton's death was arrested two days later.
"Please continue to pray for this police department," Armstrong said Sunday as news crews gathered outside the hospital where Olridge died. "It wasn't very long ago that I stood probably in very close proximity to where I'm standing now, and I said something very, very similar to what I'm saying here.
"And it just doesn't get any easier. I didn't think that we'd be here again so soon."