But officers said they are not linking the two fatalities at this stage.
They were called to an address in Esk Place at 9.48 am yesterday morning where the body of a 19-year-old man was found.
An inquiry into his death has been launched and no further information has been released.
A few hours later, at 1.30 pm, police were also called to a property in Thomas Street in the town where a 23-year-old man was found dead.
He has not yet been formally identified and enquiries are underway.
Both the men were last night being named on social media by friends as tributes poured in from around the town.
Detective Inspector Scott Young said: “Sadly, I can confirm that two young men have died in Annan.
“Police Scotland are at the early stages of investigating these deaths.
“For the deaths of these young people to occur in similar circumstances, in a short period of time and in a small geographical area, is unusual.
“We are keeping an open mind at the moment but one of the lines of enquiry is that the deaths may be drugs related so we take the opportunity to reiterate that there is no safe way to take drugs.”
Meanwhile, police have confirmed a 32-year-old woman also died in Ecclefechan on Monday.
She was found at Castle Break at 12.25 pm.
Her death is also said to be unexplained.