It happened in Uddingston Road, near Bothwell Medical Centre, between 7.30 and 7.45 on Saturday morning.
The man approached the woman, aged 77, and threatened her. At this point another woman approached the bus stop and the suspect made off towards the Nature Trail underpass.
Officers are keen for this woman to contact them as she might be able to provide additional information. The victim was not physically injured, but she has been left “extremely upset” by the encounter.
The suspect is described as white, possibly in his 40s, around 5ft 5/6in in height with a stocky build. He was scruffy and unkempt with stubble and greasy, dark brown hair. He was wearing a blue bomber-style jacket.
Inspector John O’Neill said: “I would like to assure everyone that we are doing all we can to find this man. Any person who approaches elderly people for the purpose of threatening them and causing them fear and alarm requires to be traced as soon as possible.”
Anyone with information is asked to call 101.