The body of Elisabeth-Jane Ross, 38, was discovered by a homeless man in a derelict building in the Kypseli area of the Greek capital on 18 July.
Ms Ross, from the Edinburgh area of Scotland, is believed to have died between five and seven days before she was found, police said.
It is understood she arrived in the country from the United States via Cyprus on 29 June, according to Greek media.
Earlier this week, toxicology tests were being carried out to establish whether “foul play” was involved in her death.
The Hellenic Police, Greece’s national force, were able to identify the victim using fingerprints.
In the following days after her death, police say her bank card was still being used to withdraw money and text messages were being sent from her phone.
On Sunday Greek police said a 26-year-old man, described as a foreign national, has been arrested on suspicion of manslaughter, robbery and violation of weapons laws.
The Foreign and Commonwealth Office said earlier it was “providing consular support and are in contact with the local authorities”.
A Police Scotland spokesperson said: “We are aware of the death of a Scottish woman in Athens and are liaising with authorities in Greece.
“Officers are providing support to her family in Scotland.”
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