It comes after four cases of the new variant were found in Scotland last week, among a total of 103 across the UK.
Otherwise known as B1617, it has been labelled a “variant under investigation” rather than the more serious “variant of concern”.
Chief medical officer Gregor Smith said he is more worried about the “South Africa” variant, which has become embedded in much of Europe and of which there are now 32 cases in Scotland.
“If I’m honest, although we're keeping an eye on this Indian variant, 1617, it’s not the one which I think is of most concern globally,” he told a media briefing on Thursday.
"I think particularly for us in the UK, the one which we are looking at particularly closely is the South African variant, 1351, which is now seeded widely across continental Europe and Western Europe.
“We continue to ensure that if there are any cases of those which we have identified that we respond quickly to those.”
He added: “Of the designated variants of concern that are active globally just now, that's the one that certainly I'm keeping the closest eye on, in terms of how it’s changing in its numbers.”
National Clinical Director Jason Leitch said he is more worried about the situation in India than the cases of variant which have made it to Scotland.
India is struggling to cope with a huge surge in cases which has overwhelmed health services. As of yesterday 300,000 cases had been reported, and 2,000 deaths.
"The most important thing about India in the news is not the five cases in Scotland, though they are of course important, and we continue to monitor them.
"But the situation in India is horrifying, absolutely horrifying.”
Professor Leitch said the situation in India is a reminder that Scots should not “go crazy” when restrictions ease on Monday.
He added: “There were five and a half million cases in the last seven days, and the world has passed three million deaths,” he said.
"This pandemic is far from over, globally. We have done really well at the end of this second wave, but the virus is still the virus.”