The application, put forward by Waverley Housing Association, will see the demolition of existing properties on the north of Beech Avenue, to the east of Laurel Grove and west of Larch Grove. The flats will be replaced by 109 homes.
Galashiels councillor Andy Anderson said: "It's an area that has been getting a bit tired.
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"We don't want to replace an eyesore with another one. What we have is a lovely modern take on what was there."
Mr Anderson, of the SNP, added: "There's a very active community there. I think once they start decanting people from there, quite a few of them want to move back."
Residents of the flats set to be demolished will be offered a place in one of the new properties, developers said.
The development consists of 14 blocks of terraced houses which "include a range of townhouse, terraced house and cottage flats".
Developers said the project will result in the felling of up to 79 trees – a move described as "regrettable" by SBC's principal planning officer, Barry Fotheringham.
