Ms Sturgeon particularly urged those in postcodes G41 and G42 to get tested given the “significant outbreak” in the south of the city.
The G41 postcode covers Kenmure Street, scene of the mass peaceful protest againt the attempted detainment of two men by immigration officers on Thursday morning.
She said: “The Southside has a significant outbreak just now so please take care and follow all health advice.”
Glasgow and Moray will stay in level three restrictions as the rest of mainland Scotland progresses to level two on Monday.
Case rates in Glasgow have overtaken Moray, with 80 cases per 100,000 people in Glasgow compared to 69 in Moray.
According to the latest figures cases are highest in southside areas, including Pollokshields and Govanhill.
Ms Sturgeon said the Indian variant was an “additional and very significant” factor in the decision to continue restrictions in the city.
Glasgow City Council has confirmed people can get home flow testing kits from Govanhill Community Building Car Park, Glasgow Central Mosque and Pollokshields overflow car park.
Both Glasgow and Moray will remain in level thre for at least a week, with people advised not to travel in or out of either area. A review will be held at the end of next week.
Andrew McRae, the Federation of Small Businesses Scotland policy chairman, said the news was “crushing” for businesses and their customers in both area.