Jeane Freeman said she favours digital certificates over paper versions as she believes the latter would place an unnecessary burden on the health service.
Her comments came ahead of a Downing Street news conference later on Monday at which the Prime Minister is expected to set out further details on the planned certification scheme in England.
Proposals were announced at the weekend for a "Covid status certification" scheme - dubbed "vaccine passports" - for mass gatherings south of the border, from sporting events to nightclubs.
The UK Government has said the certificates could be a mobile phone app or a paper document and they are expected to show whether an individual has received the vaccine, has recently tested negative for the virus, or has "natural immunity" having tested positive in the previous six months.
It is expected that in less than two weeks, everyone in the top Joint Committee on Vaccination and Immunisation (JCVI) priority groups will have been offered a first dose of vaccine, including all over-50s, unpaid carers and those with underlying health conditions.
Ms Freeman told BBC Radio Scotland s Good Morning Scotland programme there are ethical and equality questions over vaccine passports as not everyone can be inoculated.
She also said there are questions over how the scheme would work.
Ms Freeman said: "We're currently looking at what would be the digital infrastructure you would need for any form of certification, as we work through those ethical and equality and practical questions about how it might be used and in what circumstances.
"I don't want it to be paper - where it's possible I'd want it to be digitally done. I don't want to put an unnecessary burden on our health service - on our GP practices, for example - with everyone going to them to get the bit of paper that says 'Yes, I've been vaccinated'."
Scottish Labour leader Anas Sarwar welcomed the idea of vaccine passports but said the international aspect needs to be prioritised.
He told the PA news agency: "We need to separate the two parts around the vaccine passport discussion, one is what happens domestically and I think it would be good to have some kind of certification or card that recognises someone has had their first dose, recognises someone has had their second dose.
"It could also be used to promote it on social media to encourage uptake of the vaccine but I think there's a much more important conversation to be had about international travel.
"As people start to come to Scotland to do tourism, to do trade, to visit family, we may want them to have certification that proves they've had their vaccine, the exact same way when Scots go abroad either to holiday or to work or to build relationships in trade they might be asked by other countries to provide certification and evidence that they've had a vaccine so I think that international aspect needs to be really prioritised."