Gilmour, 20, was named Star of the Match for his performance in Scotland's 0-0 draw with England at Wembley on Friday but will now miss their key match against Croatia at Hampden Park on Tuesday.
"The Scottish FA can confirm that a member of the Scotland National Team playing squad, Billy Gilmour, has tested positive for COVID-19," the statement read.
"Having liaised with Public Health England since the positive test was recorded, Billy will now self-isolate for 10 days and therefore miss tomorrow's UEFA EURO 2020 Group D match against Croatia at Hampden."
Later on Monday, the England FA said Ben Chilwell and Mason Mount are self-isolating after coming into contact with Gilmour. The three players are Chelsea teammates and were seen embracing after that match.
The time frame means Gilmour would also miss their possible round of 16 tie, if Scotland beat Croatia and manage to qualify for the knockout stages.
The news will come as a huge blow for Scotland's Euro 2020 hopes. Gilmour was superb against England, in what was his first international start.
He drew widespread praise for his performance, with ex-Scotland international Graeme Souness labelling Gilmour "the best player on the pitch."
With Gilmour now absent, Scotland manager Steve Clarke will have to reconfigure the midfield for Tuesday's key match against Croatia. He could opt to move Scott McTominay into midfield, having started at centre-half against England, or start Southampton's Stuart Armstrong.
Speaking to reporters on Monday, Clarke said said other players now had a chance to make a name for themselves as his side look to reach the knockout stages of a major tournament for the first time.
"The only spanner that it throws in the works is that Billy would have started the game and now he won't so it is a chance for someone else to come into the team and make themselves a national hero," Clarke said.
Scotland have yet to find the back of the net in their two matches but Clarke hopes that with a bit of luck the goal drought will end.
"If we keep creating the chances we have created and keep having the shots at goal that we have had in previous matches then you would like to think that lady luck will be on our side and one of them will hit the back of the net," he said.
"If we reach the performance levels that we reached the other night [against England] I think we will be very competitive in the game," he said.
"Then you need that little stroke of luck that every team needs in a big game and hopefully it is our turn to be the lucky team."
Gilmour is rated as one of the top young prospects in the Premier League. He made 12 appearances for Chelsea last season and was named on the bench for their Champions League final win over Manchester City.
Reuters contributed to this report.