Around 400 people a day who have had both doses of the coronavirus vaccine are testing positive across the United Kingdom.
Sir David King, former chief scientific adviser to the UK Government and chair of the Independent Sage Group, said the current Covid-19 figures are "evidence of another wave appearing".
He told Sky News earlier today: "We know that anyone vaccinated twice is relatively safe against the virus.
"But let's not forget the one in 25 new cases are people who have been vaccinated twice - that means 400 new cases a day are people who had the vaccine twice.
"While there is an extensive amount of virus out there in the country, amongst our people, it is dangerous."
He continued: "Dying isn't the sole issue about that we're trying to avoid here. The number of people who are suffering from long Covid in the country is enormous and this is not a simply a flu, once you've had the vaccine."
Coronavirus cases have increased in parts of Scotland over the last month due to the spread of the so-called Indian variant.
Due to the rise in cases a number of council areas in the central belt have been kept at Level 2 restrictions.
Glasgow moved from Level 3 to Level 2 on Saturday after a number of weeks under stricter lockdown.
Sir David said: "(There are) 5,300 new cases of the disease per day in the United Kingdom and we're up about 2,000 on last week.
"Now we've been discussing whether or not we're going into a serious third wave and I don't think we can possibly wait any longer.
"This is the evidence of another wave appearing."
He has also urged the rollout of the vaccine to over-12s "quickly.
"The Pfizer vaccine has already been given the green light in this country to over 12-year-olds. I think we should run that programme forward quickly."
