Mount and Gilmour were seen embracing Gilmour at full-time and their conversation is believed to have continued into the tunnel, thereby crossing the threshold of what Public Health England deemed as a "close contact."
A statement from the Football Association read: "We can confirm that Ben Chilwell and Mason Mount must isolate up to and including next Monday [28 June]. This decision has been taken in consultation with Public Health England.
"The pair were confirmed overnight as close contacts of Scotland's Billy Gilmour after his positive test following last Friday's match.
"Chilwell and Mount will isolate and train individually in private areas at England's training base St. George's Park, with the rest of the squad returning there after tonight's fixture against Czech Republic at Wembley.
"We will continue to follow all COVID-19 protocols and the UEFA testing regime, while remaining in close contact with PHE.
"The entire England squad and staff had lateral flow tests on Monday and all were again negative, as was the case with Sunday's UEFA prematch PCR tests. Further tests will be carried out as and when appropriate."
England are already assured a place in the round of 16 in a worst-case scenario as one of the best third-placed teams but can top the group with victory over the Czech Republic at Wembley.
Should they do so, Gareth Southgate's side would return to Wembley on June 29 for the round-of-16 match, meaning Mount and Chilwell would be eligible.
However, a second-placed finish would see them miss out, as the Group D runners-up are due to play in Copenhagen on Monday, the final day of their isolation period.
A third-place finish is more complicated given the various permutations but it is possible they could feature if England end up playing the winners of Group E in Glasgow as that game takes place on June 29.
Mount's absence in particular is a significant blow to Southgate given he started England's opening two matches. Chilwell failed to make the 23-man squad against Croatia and was an unused substitute versus Scotland.
Southgate was forced to delay naming his team until Tuesday morning as discussions continued with Public Health England over whether Mount and Chilwell could play given the entire squad had tested negative.
"We had worked with the players this morning on the training pitch and then you find out as you finish that this is the situation so I can't say it isn't disruptive," Southgate said on Monday before casting doubt on how Scotland had managed to avoid any close contacts of their own.
"Look, I don't want to cause a drama for Scotland, but if you are all in the dressing room together, where does everything stand?"
The Scottish FA have insisted that Public Health England determined there were no close contacts to Gilmour in the Scotland group.
In an unrelated incident, England have also sacked the security team responsible for protecting the team away at their London hotel base before matches amid safety concerns.
The UEFA-appointed firm G4S faced complaints over allowing around 50 supporters to congregate near the team bus as they arrived at in the English capital on June 12, the night before facing Croatia.
The team's separate security detail stepped in to disperse the crowd, whose gathering is believed to have made several players feel unsafe while also breaking the government's COVID-19 rules regarding crowds.