The figures were published after UK Health Security Agency chief executive Dr Jenny Harries warned the strain was “probably the most significant threat” since the start of the pandemic.
It is the highest figure announced since mass testing began in summer last year, and surpasses the previous record of January 8 when 68,053 new cases were reported.
And it comes as an expert warned meeting with three households is “not safe”, following the publication of Scottish Government guidance.
Professor Sally Bloomfield, honorary professor at the London School of Hygiene and Tropical Medicine, said while the Government’s new guidance was “appropriate”, the public must not forget that gatherings of any size constitute a risk of Covid spread.
The Scottish Government will aim to avoid this as much as possible, Mr Swinney told the BBC.
There were 4,252 possible cases of the Omicron variant in Scotland as of December 12, Public Health Scotland has reported.
This includes confirmed and suspected cases.
Cases are more than ten times higher than the previous week and PHS estimates a doubling time of just over two days in Scotland.
Some 1,285 possible cases have been recorded in NHS Greater Glasgow and Clyde, while 956 have been reported in Lothian.
A handful of cases have been identified in Orkney, Shetland and the Western Isles.
Prof Bloomfield said mixing with fewer than three households at a time will help reduce the virus spread and may avoid the need for lockdown in January.
Responding to the new Scottish Government guidance, she said: “We have to understand the basics, which is that the virus is spread from person to person, which means the more different people we meet with, the greater the chance we will mix with someone who is infected and infectious – and become infected ourselves; and equally important, then pass it on to yet another person.”
Prof Bloomfield added: “We have a tendency to assume that if the Government tells us that it is OK for us to mix with three households, that means it is safe for us to mix with three households.
“We fail to realise that it is not safe at all. We are taking a calculated risk whenever we meet with another person.
“But the Government also realise that we have a human need to take this risk and meet with people. So, three households would seem appropriate guidance – in that it allows us to the option to spend time with those who are really important to us, or who need our care and companionship at this time, which is hard for a lot of people.
"If it works out at less than three households, this will help to reduce the spread and avoid need for lockdowns in January, which is equally important to us and the NHS.”
Prof James Chalmers, chair of respiratory research at Dundee University, said he believed new measures may be required over the festive period.
“The modelling looks like, in the worst-case scenario, we could have a really difficult winter and we can’t allow the health service to be overwhelmed,” he told the BBC.
“We need to be prepared and we need to prepare the public that we may require further restrictions.”