Speaking at the Scottish Government’s daily briefing, the national clinical director also joked that Downing Street did not want Scots to travel to the game.
The two sides are set to clash on June 18, 2021.
In answer to a question from a reporter on whether there was any hope that Scottish football fans would be able to cheer on their team in person, the national clinical director said: “The serious answer is we of course can't predict what the crowd will be in that second week in June.
“The flippant answer is that the UK Government chose June 21 because that’s three days after the game, and they don’t want the Tartan Army to travel - but that headline is not what I seek you to write.”
On Sunday, Downing Street announced that it aimed to lift almost all restrictions currently in place by June 21 at the earliest.
He added: “I am desperate to go. I really hope that, by then, travel will be allowed, crowds will be allowed - and vaccinations and studying the new variant is how we get to that stage.”
First Minister Nicola Sturgeon, who labelled Prof Leitch her “resident football expert”, said she could “not give a 100 per cent certain cast-iron guarantee.”
She added: “I certainly hope we can be in a position where we can be doing that, and I hope we come away from Wembley feeling really upbeat and positive as well.”