The town centre incidents happened earlier this month, but were just revealed by police on Wednesday.
At 5.15 pm on Tuesday November 3 a 40-year-old woman was approached by a young man near the Spar shop in the Whitesands.
He made indecent suggestions to her, but when she did not respond he walked off.
The man is described as being 17-18 years old, about five foot five tall, of average build with short dark hair. He was wearing a white coloured top, black Adidas jogging bottoms and blue trainers and he spoke with a local, polite accent.
The second incident happened a week later at 6.30 pm on Wednesday November 11 in Terregles Street at the junction with King Street.
On this occasion a 25-year-old woman was approached by a man who made indecent sexual suggestions to her. He is described as being in his early 20s, five foot two tall, slim build, short dark hair and was wearing a grey hooded top with dark coloured jogging bottoms.
Neither of the women were injured but they were left shocked.
Constable Victoria Urwin at Dumfries said they are unsure if the two incidents are linked.
She said: “Whilst some of the details of the suggestions are similar in content, the descriptions of the men do vary, and as such we are keeping an open mind.
“We are interested in hearing from any other women who may have had a similar approach, or indeed anyone who may be able to help us identify the man, or men responsible.”
Officers can be contacted on 101.