The reports did not identify the man and did not provide any detail.
Italian officials have not announced the arrest and CNN has not independently confirmed it.
Previously on Wednesday, a prosecutor and lead investigator in the case told media that police know the identity of a suspect.
It's not clear whether the man reported arrested was the same person mentioned earlier.
Investigators were able to identify the suspect in the image, taken from closed-circuit surveillance footage, investigator Domenico Giacinto Profazio said during an impromptu news conference. Prosecutor Giuseppe Creazzo told journalists the same thing.
They did not disclose the suspect's name.
No further information on the image's origin was immediately available. Investigators have said they've reviewed hundreds of hours of footage from multiple cameras.
Olsen was found dead -- apparently strangled with a rope or cord, according to Italian media reports -- in her apartment on Saturday. Creazzo told reporters only that it appeared Olsen was strangled with something other than bare hands.
The woman was not sexually assaulted, and exams meant to determine whether she had had sex before her death were not available, he told reporters. Toxicology reports, he said, could take weeks to be completed and may remain sealed during the investigation.
Olsen's boyfriend, Federico Fiorentini, 42, had been unable to reach her Saturday and got Olsen's landlord, Claudia Colivicchi, who lived upstairs, to open the apartment for him, Profazio told CNN in a phone interview.
Colivicchi told Italian media outlets that when she opened the door, Olsen was on a bed in a mezzanine loft of the studio apartment. Fiorentini moved her to the floor and performed CPR, she said. Creazzo confirmed in an interview with CNN that Fiorentini tried to revive her.
Olsen's dog, Scout, who was a regular feature on her Instagram and Facebook feeds, had defecated in multiple places within the apartment, indicating that the well-trained beagle had not been taken for a walk in some time, Colivicchi told media outlets.
Fiorentini has an alibi that lines up with testimony from other witnesses, Creazzo said earlier this week. But, the prosecutor said, "no one has been excluded" as a suspect in the case.
Olsen's body was removed from the apartment Saturday night, and the autopsy was completed Monday. Forensic investigators returned to Olsen's home Wednesday to gather more evidence.
The woman's friends told investigators they were with Olsen at the Montecarla nightclub in Florence on Thursday night and left her there between 3 a.m. and 4 a.m. Friday, local media reported.
The American's phone was turned off at about 9 a.m. Friday, police said, adding that they have not found her phone.
Authorities are looking into a photo, posted on Olsen's Instagram feed in October 2015, which carried the caption, "I have a stalker." The picture is taken from above street level and appears to show Olsen, in jeans and a tank top with a cup in hand, and Scout walking down the street.
It's unclear whether the post was serious, as it carries numerous hashtags, including #stalkeralert, #creeperpic and #scoutthebeagle.
Olsen was originally from Summer Haven, Florida. She was living in Florence, where her father, an architect, works at an art school. Her father was seen with friends outside her building Monday, leaving flowers.
"The police haven't really said anything yet. My son is waiting for the police to share more," Olsen's grandmother, Ann Olsen, told CNN by phone from Florida.
Olsen's friends told CNN affiliate WJAX in Jacksonville, Florida, that Olsen was "full of life."
"She never met a person who didn't like her. ... She had the best personality," Staci Kelly told the station, adding that Olsen loved to travel. "She never really stayed in one spot for too long."