The development will include a four-storey block containing 30 flats for elderly residents, along with a row of three bungalows.
A three-storey block with 12 flats for adults with learning disabilities will also be built. The council will offer all the homes at social rent.
Murray House shut in 2018, after the care watchdog found in 2017 that the number of people permanently living at the care home on Acre Road had decreased.
