Chelsea midfielder Gilmour embraced his club teammates after England and Scotland's goalless draw last week.
Gilmour has since tested positive for COVID-19, leaving Mount and Chilwell in danger of catching the virus.
Asked if the pair will be fit for Tuesday's game, Southgate replied: "We don't know at the moment.
"There's obviously got to be quite a doubt but there's still a lot of discussions and investigations going on behind the scenes, so at the moment they're isolating and we just have to find out over the next 12 hours or so.
"We had worked with the players this morning on the training pitch, so of course then you find out as you finish that this is the situation, so I can't say it isn't disruptive of course.
"We don't really know at this point whether they might be ok for tomorrow or they could be out for 10 days, so there's a lot of unknowns frankly at this moment in time.
"Although we worked on the training pitch, we hadn't confirmed the team. We put some players in and moved them around, so we hadn't confirmed the team yet and we'll wait until the morning when we're really clear on the final situation on the two boys as to where we stand."