The man followed her from a Glasgow social club to her house in Carntyne Road, in the East End, and then posed as a "water board" worker to gain access to the pensioner's home.
But the woman soon became suspicious and started shouting at him.
The conman took fright and quickly ran out, leaving empty-handed.
Police called the incident, which happened about 3.15pm on March 19, "deplorable".
And they warned pensioners to be on their guard against bogus callers after a spate of cruel thefts across the city.
Detective Constable Cameron MacLean, from London Road CID, is leading the hunt for the conman.
He said: "This was a frightening experience for the woman involved, who had just returned to her home from a local club.
"We are working on a possibility this male followed her before he went to her address.
"To target elderly and vulnerable people in their home is deplorable."
The man was described as aged 20-35, white, and about 5ft 4in. He was of medium build and clean shaven.
He was wearing a black jacket, dark trousers and black shoes.
DC MacLean added: "We would like to speak to anyone who may have seen a male matching this description hanging around Carntyne Road, just before or after the incident happened.
"Anyone with information is asked to contact police.
"Unfortunately, bogus caller type crime can be quite common, so we would like to remind residents to be vigilant, particularly where elderly neighbours or friends are concerned."
As previously reported by the Evening Times, conmen have targeted several elderly people during the last six months.
All cases have involved a caller claiming to be from the water board.
During one week in October, five elderly people on the South Side were targeted by bogus callers.
In January, crooks stolen hundreds of pounds from an 89-year-old woman in Ashmore Road, Cathcart.
Anyone with information on the Carntyne Road incident is asked to call London Road police station on 0141 532 4600.