The flats are designed as amenity homes for social housing, with disability access, and the buildings will be fitted with lifts.
They will be constructed in three blocks of six flats, and work is expected to start in November.
The new homes could be ready by September next year.
The plans attracted one letter of objection.
When the baths were demolished, having been replaced by a pool in the Lanark Lifestyles complex, its car park remained in use, but parking spaces created as part of the new plan will be for the flats’ occupants only.
The objector stated that the removal of the existing car park would be to the detriment of residents as there is no on-street parking, use of the nearby Bernard’s Wynd car park is restricted, and the Wellgatehead car park to the south of South Vennel is seemingly not suitable “for security reasons”.
However, members of the planning committee were told that use of the former baths car park was on an informal basis and that there was no requirement for the council to provide parking for existing private homes without parking spaces.
Clydesdale North councillor Catherine McClymont was pleased to see the amenity homes approved.
“There is a healthy demand for this type of housing in Lanark, and we are hoping that this will alleviate the waiting list in the area,” she said.
“The council under the previous Labour administration committed to this development. It has left a legacy to ensure that the council fulfils its pledge to build 1,000 new homes in South Lanarkshire.
“I look forward to more homes in my area”