A woman in her 80s had her handbag stolen after two men came into her home in Sherwood Drive, Verwood on Tuesday 3 November.
She opened her door to a young man at around 7.30pm after he claimed to be working at a nearby property.
A second man arrived and they both entered the house. While inside, the offenders stole her handbag.
They left the scene when they realised she had activated her Careline alarm.
Earlier, at around 6.30pm, a young man had knocked on the door of an address in Owls Road claiming he was looking for someone.
Shortly after, an older man went to the back door and knocked persistently.
The householder, a woman aged in her 70s, refused to open the door and the two men left.
Neither victim was physically injured in the incident.
DC Christian Bryant, of Bournemouth CID, said: “Unfortunately neither victim has been able to give us a description of the offenders but we do believe the same two men were responsible for both incidents.
“I am appealing to anyone who saw two men acting suspiciously in the area at the time of the two incidents to please call me.”
No arrests have been made.
Witnesses and anyone with information should call Dorset Police in confidence on 101 quoting incident number 3:342. Alternatively, call the free and anonymous Crimestoppers line on 0800 555 111 where mobile phone tariffs may apply.