"There is still a lot of discussions and investigations going on behind the scenes," England coach Gareth Southgate said. "At the moment they are isolating and we need to find out over the next 12 hours or so."
Gilmour has already been ruled out of Scotland's game on Tuesday after entering isolation for 10 days.
"He's upset as you would expect," Scotland coach Steve Clarke said. "He's asymptomatic. Hopefully he'll get back to playing as quickly as possible. It's a blow for us. But it's a chance for someone else to come into the team."
Gilmour was UEFA's player of the match on Friday in the 0-0 draw.
Players are regularly tested and are staying within team bubbles except when they are at matches. The Scottish Football Association said it had been liaising with Public Health England, but did not announce if anyone else had been deemed a close contact and would have to also isolate.
Clarke refused to answer questions about how the bubble was breached or how Gilmour may have come into contact with the virus.
"That's not my department. That's for the medical department," Clarke said. "I'm just focusing on preparing the team for tomorrow's match."
It was the second time in less than a month that the coronavirus has hit the Scotland team. Midfielder John Fleck tested positive on June 1, and six other players were told to self-isolate and were unavailable for a friendly against the Netherlands.
Croatia and the Czech Republic were initially scheduled to use training bases in Scotland but they canceled those plans on May 31, citing UEFA concerns about local COVID-19 rules. The fear was that if one player tested positive, other players and staff members would also have to self-isolate.
Croatia coach Zlatko Dalić reacted to the news of Gilmour's self-isolation with a measure of irritation, saying the rules weren't the same for all.
"The conditions should be the same for all," he said. "They're not the same for all. I wish for Gilmour to recover well. And I don't wish for anymore Scottish players to test positive."
Scotland's training base is in Middlesbrough, England, so the rules that Dalić is complaining about don't apply.
Dalić said he would have preferred to set up camp in Scotland, since the team is traveling back and forth to Croatia.
"We've made an exception and we stayed in Croatia," Dalić said. "And we're traveling every three days. Those are not good conditions."
The coach said he doesn't expect that any of his players will be afraid of catching the coronavirus and request not to play in Tuesday's match.
"We just arrived in Glasgow," Dalić said. "We went to have lunch. We will have a meeting. I don't expect anybody to request that."
Scotland and Croatia both have one point from their two Group D matches.
