The 30-year-old was apprehended while trying to travel to Bulgaria via Evros in northern Greece with a fake passport, the Kathimerini reported on Sunday.
The unnamed Georgian man was reportedly linked to another break-in at a house near where Ms Crouch was murdered in the Athenian suburb of Glyka Nera.
DNA analysis suggested he was involved in tying up an elderly couple during a burglary in March, various UK outlets reported a police source as saying.
The person arrested in Evros has not yet been brought to the homicide department, which is looking at the possibility that he is connected to Crouch’s murder, Naftemporiki reported.
Ms Crouch, 20, was sleeping alongside her husband, Charalambos Anagnostopoulos, 32, a pilot, and their baby when three burglars broke into their home shortly before dawn on Tuesday.
She was tied up and eventually strangled in front of her baby while her husband was bound and gagged as the intruders searched their property, making off with cash and jewellery.
Mr Anagnostopoulos managed to call police after loosening his bonds.
“The robbers were armed and they threatened to kill both the husband and the wife on separate occasions,” said Constantine Hasiotis, chief of the homicide department, who is leading the investigation.
The thieves killed the family’s dog and left it hanging on the fence of the house, police said.
Findings from the investigation showed that Ms Crouch, an avid athlete and kick boxer, initially moved to resist her attackers, Mr Hasiotis said on Wednesday. But an autopsy report from a state coroner said: “There were no bruises or signs that she may have struggled with them.”
The Greek minister responsible for public order, Michalis Chrisochoidis, described the killing as “particularly heinous”, saying: “One rarely encounters such barbarity in Greece, in Greek society, even among criminals.”
The crime has sent shockwaves across Greece and beyond, prompting the government to announce a €300,000 (£260,000) reward for any information leading to the arrest of the assailants.
And just days before Greece reopened its borders to non-EU tourists on Friday, hoping to welcome millions of Britons, the country’s justice minister said the government would stiffen legislation for convicted robbers and rapists, forcing them to serve out at least 20 of the 25 years they face.
Since the bounty was announced late on Tuesday, Mr Hasiotis said thousands of calls had been received by the authorities from people claiming to hold answers to the murder.
“But it wasn’t the reward that triggered this,” Mr Hasiotis said on Wednesday. “It was the knee-jerk reaction of a sensitised society, stunned by such a heinous crime.”
The bounty may prove pivotal in helping to entice a person on the inside with a “nugget of information that can help move this investigation further”, said Mr Hasiotis, adding: “That bit of credible information has not come in yet. But that is not stopping us from pursuing all other avenues of investigation.”
Local media reports that police believe the assailants to have histories immersed in illegality, with investigators having contacted prisons to draw up a list of notorious and recently released criminals who have served sentences for robbery.
Between 20 and 30 of them are considered “professional” criminals capable of such a crime, reports suggest.