The Covid-19 testing system has been hit by "exceptional demand" which has seen some people in Scotland only offered tests in England.
Nicola Sturgeon has said she was aware of "possible technical issues" with the UK-wide online booking system.
It has led to some people in Glasgow and the central belt being directed as far afield as Penrith in Cumbria.
The problems have been highlighted after positive coronavirus tests at schools in Glasgow and Lanarkshire.
BBC Scotland has also been contacted by Glasgow parents who said the closest test centre made available to them was in Stranraer.
They would ordinarily expect to be offered an appointment at Glasgow Airport.
The first minister said the government was "looking right now at how the demand is properly and appropriately managed".
She added: "There may have been some technical issues with some people in Scotland, instead of being directed to the next available slot in Scotland, being directed erroneously to testing centres in England.
"We are working to ensure that is resolved.
"I would say to people: our testing system is obviously working and it's working well. But at periods of high demand, bear with us as we resolve any particular issues."
The UK government's Department of Health and Social Care confirmed tests had been fully booked at some sites - but that it should be a short-term problem.
A spokesman said: "NHS Test and Trace is working and testing centres are open across Scotland for people to book a test if they need one.
"Appointments at some sites have already been filled today, but more will be available to book again for tomorrow and home testing kits can still be ordered.
"Anyone with symptoms across the UK should get a test as soon as possible as well as washing hands regularly, wearing face coverings and following social distancing rules to stop the spread of the virus."
Scottish Labour's Monica Lennon said the problem was part of a wider issue over testing.
She said: "We've had months to boost capacity and testing and we're still not getting it right.
"It is, frankly, astonishing that people in Glasgow are having to travel over the border to England for a Covid-19 test.
"Something has gone badly wrong and the Scottish government has to fix it rapidly."
Scottish Conservative health spokesperson Donald Cameron said: "I urge the Scottish government to find out what's gone wrong here and work quickly to maintain public confidence in the testing system."