No close contacts in the rest of the squad were identified by Public Health England meaning Steve Clarke won’t be missing any other players.
England put Ben Chilwell and Mason Mount, Chelsea team-mates of Gilmour, into self-isolation as a precaution ahead of their Euro 2020 match with the Czech Republic.
The trio embraced and talked for a sustained period after the 0-0 draw between the countries on Friday night.
Southgate, while not wanting to cause any issues with the Scotland camp, was left wondering about close contacts with Gilmour having been in the dressing room with team-mates.
He said: “I don’t want to cause a drama for Scotland, but if you are all in the dressing room together, where does everything stand?
"I really don’t know how all that works. All I know is we are in the situation and we’re complying with Public Health England.
“There’s nothing else we can do. I don’t want to make life difficult for Steve Clarke. I want him to do well. He’s a great guy and we’ve got a very good relationship.
“The last thing I want to do is make his life more difficult, frankly.”
England have already secured qualification no matter the result against the Czechs at Wembley tonight, while Scotland will join them in the last-16 with a win over Croatia.