The latest data shows there are 58.3 infections per 100,000 population and a senior health official has warned that there are clusters of high infection rates in parts of the city.
It comes as all local authority areas on the mainland except Moray must move to level two of the route map outside of lockdown Monday.
The Prime Minister tweeted: “We are seeing an increase in cases in Glasgow right now and, amid concerns about the so-called Indian variant, we are watching it very closely. “
It followed comments from Linda de Caestecker, director of public health at the NHS Greater Glasgow and Clyde, that an increase in the number of cases was to be expected with the easing of restrictions, but she was “concerned” about an increase in Glasgow.
She told Radio Scotland: “The numbers are increasing very, very quickly – especially as we see new, more transmissible variants coming in.
“What we do know is it’s not the Kent variant, so we assume it’s the Indian variant, but we don’t have the sequencing yet so we can’t be sure. “
Meanwhile, weekly deaths from Covid-19 in Scotland have fallen to single figures for the first time since September last year, according to the latest data.
Figures from the National Records of Scotland (NRS) show that seven coronavirus-related deaths were recorded between May 3 and 9, down 12 from the previous week.
This is the lowest total in nearly eight months.
Deaths from the virus have been steadily declining since mid-January, when they hit 452 during the second winter peak.
They were the last in single digits in the week starting September 7, when five deaths were recorded.
The latest data shows 10,104 people have now died from the coronavirus in Scotland since the start of the pandemic.
In the most recent week, North Lanarkshire recorded two coronavirus-related deaths, and Angus, Glasgow, Inverclyde, Midlothian and South Lanarkshire all recorded one.
Five deaths occurred in hospitals, one in a nursing home and one in a non-institutional setting such as at home.
Pete Whitehouse, director of statistical services at the NRS, said: “There were seven Covid-related deaths recorded last week, which is the lowest number since September 2020.
“Across Scotland, no Covid-related deaths have been recorded in 26 of 32 local communities.
“The total number of deaths from all causes was 12% below the average level for this time of year, although last week included a public holiday and some death registrations may have been delayed.
The statistics are published weekly and cover all registered deaths in Scotland where Covid-19 is mentioned on the death certificate.
They differ from the laboratory-confirmed coronavirus deaths reported daily by the Scottish government as the NRS figures include suspected or probable cases of Covid-19.
As part of the daily measurement, 345 new cases of coronavirus have been recorded in the past 24 hours, and no deaths.