They were confronted by a gunman and the officer was shot. He was taken to Royal Shrewsbury hospital but pronounced dead shortly after.
A man in his 30s, yet to be identified, was found dead at the property in New Park Road and a gun was recovered.
A spokesman for West Mercia Police said: "Officers were called to an address in New Park Road, Shrewsbury… this morning.
"We can confirm that two people have died in this incident. The Independent Police Complaints Commission has been informed."
Residents living near the suburban street reported hearing two loud bangs at around 06:00 BST.
Tony Robinson, relief manager at a newsagent, said: "I heard some shouting then a couple of loud bangs.
"I looked out of the window and there were armed police swarming all over the place.
"No-one really knows what has happened."
Jan Almark, whose husband runs a building yard near the scene, said the shooting took place in a flat development.
She said: "My husband was going to work this morning but the police stopped him at the end of the road and said he couldn't go through.
"He was told two people had been shot but the police were being very tight-lipped about what had happened.
"It happened in a set of flats next to a car park for the shopping centre."
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