The Scotland squad have been based at Middlesbrough’s Rockliffe Training Ground during the tournament but the Scottish FA currently say no close contacts have been identified from inside the camp.
"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."
Scotland held England to a goalless draw at Wembley last week, yet match play is not considered close contact.
England's players and support team all returned negative tests on Sunday after Friday’s match against the Scots.
Scotland must beat Croatia at Hampden Park to stand a chance of qualifying for the knockout stages.