Prof Leitch said the decision to keep Glasgow in level three restrictions went “down to the wire” and indicated the city may have to stay at the current level for longer than a week.
With the exception of Glasgow and Moray, which are dealing with Covid-19 outbreaks, all mainland areas will move from level three to level two restrictions on Monday.
Changes to the rules include the ability to hug loved ones again and indoor socialising in homes.
Many island communities will move to level one.
Nicola Sturgeon announced on Friday that Glasgow and Moray would have to stay at level three, with the situation being reviewed at the end of next week.
Speaking to the BBC’s Sunday Show, Prof Leitch said initial signs suggested the Indian variant was “part of the challenge” in Glasgow.
Asked if the outbreaks were likely to spread to other areas, he said: “I think we should watch and be cautious and careful.
“We have decided, with advice, that on Monday the rest of the country is safe to move to the next stage.
“But we all said the hugging, the increased hospitality, that should all be done very cautiously.
“So don’t go crazy, but we think it can be kept under control.”
Prof Leitch said it “may well be” the case that Glasgow stays in level three for longer than a week
On Monday in the level two areas, up to six people from three households can socialise indoors in a private home or garden without physical distancing.
Physical contact such as hugs with loved ones will be allowed again, though the government says people should use their judgment around how often this takes place.
Pubs can open and sell alcohol indoors until 10:30pm in two-hour booked slots. Local licensing laws will apply outdoors.
On Sunday, a total of 292 cases of coronavirus and no new deaths were recorded in the Scottish Government’s daily figures.
A total of 3,020,335 people have received the first dose of a Covid vaccination and 1,621,031 have received their second dose.