Newly released bodycam footage and transcripts show the man, Sean Bickings, 34, getting into Tempe Town Lake last month and warning police he was "going to drown".
But none of the officers on the scene intervened to save him, with one heard saying: "I'm not jumping in after you."
Mr Bickings soon slipped underwater and was later declared dead.
Describing him as "an unsheltered Tempe community member", city authorities called his death a "tragedy".
The incident took place after 05:00 local time (12:00 GMT) on 28 May, when officers responded to a call about an alleged fight between the man and his partner in the city centre.
According to the city's statement, the couple "cooperated fully and denied that any physical argument had taken place".
However, police footage showed that while officers were checking the couple's names for outstanding arrest warrants, Mr Bickings climbed over a metal fence dividing the boardwalk from the lake and entered the water.
He told the officers he was going "for a swim", waded in and swam towards a bridge even after they informed him swimming was not allowed in the lake.
The 12-minute bodycam video ends around this time, with a message at the end saying it has been cut short "due to the sensitive nature of the remaining portion of the recording".
The city instead provided a transcript of the remainder of the exchange, in which Mr Bickings repeatedly said he was on the verge of drowning and needed to be saved.
The transcript also showed officers tried to calm down Mr Bickings' companion as she grew increasingly distraught, telling her they were getting a boat.
At one point, she shouted: "He's drowning right in front of you and you won't help."
Mr Bickings did not resurface.
The three officers involved in the incident have not been named.
They are on "non-disciplinary paid administrative leave" as their response is examined by the Arizona Department of Public Safety and the police department of neighbouring Scottsdale.
Photo Tempe City Caption: Police body cam footage shows the officers interacting with Sean Bickings before he gets into Tempe Town Lake