Detectives believe the 37-year-old woman was conducting business with a man in the midlands town and suspect she may have been killed by this person.
Ms Greene disappeared just over a week ago from her home on St Mel’s Terrace, where she lived with her mother and sister.
In a subsequent appeal for information, her mother, Winnie, said she feared her daughter had either been murdered or abducted.
Gardaí have not recovered her body, but received information following the appeal leading them to suspect she may have been killed.
It is understood gardaí made a breakthrough after checking who Ms Greene was in phone contact with before she left her house. Her mother said she got a phone call just before she left, at 9pm last Sunday week.
Detectives dug deeper into the connection between Ms Greene and the caller and have developed a good picture as to what might have happened.
As a result, they moved in on a house in Monksland, on the western part of the town, early yesterday morning. They arrested a man, aged in his early 60s. Garda sources said he did not have a criminal background.
The man was detained under Section 4 of the Criminal Justice Act, which allows gardaí to hold him for 24 hours, excluding breaks. He is being questioned at Athlone Garda Station.
Gardaí confirmed that he was arrested on suspicion of murder. Garda sources suspect her body may have been buried locally. They have not ruled out the possibility that she may be alive.
A massive search operation was put in place over the weekend, with the Garda Air Corps and the Garda Sub-Aqua Unit brought in to aid in the search.
The Garda underwater unit began searching parts of the Shannon on Saturday and this continued up to yesterday. Gardaí are also searching fields and bogland, assisted by the garda helicopter.
The house at Monksland was sealed off pending the arrival of the Garda Technical Bureau from Dublin. The expert team arrived yesterday and began a detailed forensic examination of the house.
Gardaí issued an appeal for information last Tuesday for Ms Greene, who they said was last seen at around 9pm on Sunday, February 13.
She was described as around 5’5” in height, of slim build and fair complexion. Gardaí said she had medium length brown hair, normally tied up in a pony tail.
When last seen she was wearing blue jeans, white runners and a navy zipped up fleece. Her photograph was also released to the media to jog people’s memory.
As a result of the appeal, gardaí in Athlone received up to 100 leads, with reports of her being spotted in different parts of the town.
* Anyone with information should contact Athlone Garda Station at 090-6498550 or the Garda Confidential Phone Line at 1800 666 111.