The incident Monday occurred in Mirzapur village in Jhalakathi district, 124 miles south of the capital of Dhaka, the BBC reported quoting police.
Police said the woman identified as Monju Begun, a mother of three, brought the severed organ to the police station for evidence.
The woman was quoted as telling police the alleged attacker forced his way into her makeshift home and started to assault her.
The victim, who was admitted to a nearby hospital, denied her accusations, the report said.
The woman told police the man, a father of five children, had been harassing her for months.
From his hospital bed, the man told the BBC he had been having an affair with her and claimed she had suggested leaving for Dhaka together to settle in the capital.
"I refused and told her that I cannot leave my wife and children, so she took revenge on me," he was quoted as saying.
"We will arrest him once his condition gets better," a police spokesman told the BBC.
"She said she fought back and cut off his penis and brought it to our police station in a polythene bag to prove" the man tried to rape her.
A hospital surgeon said doctors were not able to reattach the organ as it didn't arrive in time after the incident.
"We are treating him so that he can urinate normally without the penis," the surgeon told the BBC.