The Scottish Government said it was “closely monitoring” the situation, but hoped that Glasgow would move to level two restrictions as planned on Monday.
While genomic testing has yet to be completed, it is believed that at least some of the new cases are the variant of concern first identified in India, B16172. A total of 35 cases of this variant have been confirmed in Scotland, although some data suggests the figure may be higher.
The current outbreak is centred around the south of the city, in Pollokshields and Govanhill in particular, with some cases also in Easterhouse.
Over-18s in such areas will now be offered their Covid vaccine earlier than planned.
The Scottish Government said the acceleration will include “re-offering to those in eligible cohorts” as well as “deploying additional vaccine stocks to accelerate deployment in remaining cohorts in affected areas”.
NHS Greater Glasgow and Clyde is “actively pursuing” was to make sure vaccine uptake is as high as possible, a spokesperson said.
The Pollokshields East neighbourhood, which has the highest case rate per population in Scotland, at 23 times that of the the country, saw a huge crowd gather on Thursday as people protested the detention of two men by the UK Border Agency.
Police Scotland cited protection of public health as one reason for the men's release after eight hours.
It comes as NHS Grampian reported that while cases in Moray were reducing, some other local outbreaks had sprung up. Professor Linda Bauld, chair of public health at Edinburgh University, said these outbreaks were a “warning” and must be tackled immediately to stop them jeopardising Scotland’s easing of restrictions.
Glasgow currently has a seven-day case rate of 71 per 100,00 population - 2.5 times that of Scotland as a whole. The rate in Moray is just under 100.
Dr Linda de Caestecker, director of public health at NHS Greater Glasgow and Clyde, called the rise in cases a “cause for concern”.
“A rise in household mixing seems to be the principal reason for the increase,” she said.
"I do understand the temptation to think if we can do it on Monday, why not just do it now, but to prevent further increases in case numbers, I would urge everybody to stick to the rules as they are now, not as they are likely to be in a few days’ time.
“In addition, we are seeing new variants that are more transmissible and that is another reason why we cannot be complacent at this critical time.
“The numbers can go up very quickly, and it is the responsibility of all of us to play our part and prevent further increases. At the moment, Glasgow is due to drop to level two next week, and the more we can do to stop spread of the disease, the better.
A Scottish Government spokesperson said: “Our ambition remains that the country will be able to move together in the easing of restrictions and we are working with the National Incident Management Team to closely monitor the situation in Glasgow to ensure that any decisions best support the progress that has been made in reducing the prevalence and impact of the newly detected strain of virus in Scotland.
“The First Minister confirmed on Tuesday that all mainland council areas would move to level two on Monday, May 17 with a decision on Moray due to be made at the end of the week and that intention has not changed.
"Moves down through the levels for all areas are always subject to developments in the epidemic, however, and we are monitoring the situation in Glasgow and remain in close contact with local government and public health leaders.”
It comes just days after Nicola Sturgeon announced that Moray, which has seen a recent spike in cases, may have to remain for some time in level three.
Cases are also rising in Midlothian, which has been pinpointed as possibly the next area of local concern.
NHS Lothian declined to comment on whether or not the easing of restrictions might be at risk in the area, saying that decision could only be taken by the Scottish Government.
Prof Linda Bauld called these local outbreaks a “warning” for the rest of Scotland.
While there is currently “no strong grounds” to suggest mainland Scotland should not move to level two on Monday, Prof Bauld said if cases rise any further then the nation will need to “press pause”.
“I don't think any area is totally safe," she said.
"We're in the middle of a big experiment - opening up when we’ve only got partial protection. I'm not going to say I think everything's going to be okay.
"I think Moray and Glasgow are a warning to us, but we do not have evidence at the moment to say that we should stop moving forward, because all the indicators in most of the country are still going in the same direction.
“There is no strong grounds to say don't go ahead on Monday nationally. It's about being alert and recognising two things: That we need strong action to get on top of outbreaks, and secondly to look at any signs [of rising cases nationally]. If that keeps creeping up every day, then we do have to press pause.”
NHS Grampian said on Thursday that the Moray outbreak has seen signs of improvement, but warned that new outbreaks are now being seen in Lossiemouth, Keith and Aberlour.
“One person not self-isolating when asked, one mass gathering, people gathering inside and every other breach of the guidance, has the potential to be the start point of a new cluster, that ultimately means Moray's rates do not drop to where they are compatible with level two restrictions. So stay patient and keep sticking with it,” said deputy director of public health at NHS Grampian, Chris Littlejohn.
“Despite some positive signs over the last few days, the figures in Moray are still well ahead of the rest of the country. We are starting to see signs of outbreaks in other communities outside of Elgin, including in Keith, Lossiemouth and Aberlour."