Speaking at her lunchtime covid-19 briefing, Ms Sturgeon also said case numbers in East Renfrewshire may have fallen but that it was ‘too early to be absolutely certain’.
The news comes as it was confirmed that – as of Tuesday, May 25 – more than five million doses of vaccines have been administered in Scotland.
Ms Sturgeon said this is a ‘fantastic milestone’ in the vaccination roll out.
Speaking about the situation in Glasgow and East Renfrewshire at the covid briefing on Tuesday May 25, Nicola Sturgeon said: “I am pleased to say that there are some signs for cautious optimism here.
"Firstly it appears that the situation may have stabilised in East Renfrewshire and indeed case numbers there may have fallen very slightly.
"That said, it is too early to be absolutely certain so it remains important that residents continue to come forward for testing and vaccination and that everyone continues to exercise care.”
Despite Glasgow’s seven day average of coronavirus cases increasing from 112 per 100,000 people to 137 on Friday, Ms Sturgeon said the ‘extensive public health efforts’ will take time given the ‘scale and complexity’ of the city.
She added: “It’s important we don’t lose heart.
“In fact in the past two days or so we do think we may have started to see signs that case numbers in Glasgow are stabilising and test positivity has also remained fairly stable in the past few days at around 4% so that would be another reason to be positively optimistic.
However, talking about Scotland’s way out of lockdown, Ms Sturgeon urged that ‘significant caution’ is still necessary as Scotland records 2 deaths in the past 24 hours.
Ms Sturgeon pointed out that Scotland is facing rising case numbers of coronavirus overall at the moment.
She said: "If you look back over the past 16 days, case numbers, albeit from a very low level have actually doubled and while some of this will be down to increased testing, the test positivity rate has been creeping up as well.
The First Minister also noted that the presence of the April 02 variant was another important factor in the rise of case numbers.
However, if vaccination is successful in ‘breaking rising case numbers’, Ms Sturgeon said: "While care will still be needed, it might mean that we don’t have to react quite so aggressively with tough restrictions in the face of rising case numbers.
“Instead we hope we will be able to rely more on enhanced public health interventions like testing and vaccination and of course on good public health practices on the part of the public.”
The First Minister added that the scale and the extent of the vaccination programme will enable Scotland to ‘change the response to the virus.’
The Scottish Government is currently working to accelerate the vaccination programme which includes speeding up second doses.
A further announcements about the future of restrictions in Glasgow will be made at 12.15pm on Friday.