A man in his thirties has been arrested on suspicion of killing the pair – a man in his sixties and a woman in her seventies.
It is believed he was known to the victims.
He was detained after police were called to a property in the town’s Regent Crescent at 7.30am on Tuesday.
Detective Inspector Steve Menzies, of North Yorkshire Police’s major investigation team, said: “Local residents will understandably be shocked to hear about this tragic incident.
“Although the investigation is at an early stage, a man has been arrested, and we are not currently looking for anyone else in connection with the deaths.
“Officers from the Neighbourhood Policing Team are conducting reassurance patrols in the area, and anyone with concerns can speak to them.”
Police family liaison officers were providing support to those affected, he added.
Anyone with information that could assist the investigation is asked to contact North Yorkshire Police on 101, select option 2 and ask for the Major Investigation Team.