Gilmour was named man of the match in Friday's goalless Euro 2020 draw between England and Scotland at Wembley but will miss Tuesday's final group game against Croatia after contracting the virus.
The Chelsea midfielder's positive test sparked concerns that members of Gareth Southgate's squad could have been infected after coming into close contact with Gilmour on Friday.
The Football Association have confirmed, however, that all 26 players and members of the wider support team returned negative results after undergoing PCR tests on Sunday.
The full squad will train at Spurs Lodge on Tuesday morning, in the last session before their final Group D outing against the Czech Republic.
The England camp remain in contact with Public Health England and preparation for Tuesday's fixture at Wembley are proceeding as normal.
Gilmour's negative test is a blow to Scotland, who must beat Croatia at Hampden Park on Tuesday in order to have a chance of reaching the knock-out stages.
The 20-year-old will now self-isolate for 10 days but no other members of Steve Clarke's squad are affected as close contacts.