Nicola Sturgeon has said she still has concerns about the amount of time being taken to process coronavirus tests at UK Government laboratories.
She again spoke of pressures on the testing system in England which have caused a delay in people getting results.
Her comments came as a UK Government minister insisted coronavirus testing capacity in Scotland is “increasing enormously”.
Iain Stewart also said that if decisions need to be made over who should be the priority for testing in Scotland, that would be for the Scottish Government.
Coronavirus tests in England are to be rationed as the Government at Westminster struggles to get to grips with soaring demand.
UK Health Secretary Matt Hancock said there will be testing “prioritisation” for people with acute clinical need and those in social care settings, as he acknowledged “operational challenges” in the system.
First Minister Ms Sturgeon stressed on Tuesday there was “not by and large” an issue with getting tests north of the border.
But she said “constraints” at the UK Government’s Lighthouse laboratories meant results were being delayed – an issue she has raised in talks with Mr Hancock and Dido Harding, the head of the UK testing system.
Speaking at her coronavirus briefing on Wednesday, the First Minister said: “Although we continue to have some concerns about the time being taken to process tests in the Lighthouse laboratories, there are no signs at this stage of people in Scotland facing widespread difficulties in booking a test.”
She said the backlog in testing is now reducing and her Government will “monitor these issues very closely”.
She added she is considering if the regular testing of care home staff – which is currently dealt with by the Lighthouse laboratory system – could be taken over by the NHS in some areas.
Ms Sturgeon said: “The reason we have been looking at that is to see if we can free up capacity within the UK system, given the pressure it has been experiencing.”
Mr Stewart stressed the UK Government, which he said is responsible for the majority of testing in Scotland, is acting on the problems.
Speaking on BBC Radio Scotland, he said the Government is “increasing the overall capacity” in the testing system.
He insisted it is right that the UK Government had acted to prioritise access for testing in England to key workers and those in care homes.
Asked what should happen in Scotland, Mr Stewart said: “That is for the Scottish Government to determine.
“What we are doing as a UK Government is increasing enormously the availability of testing in Scotland.
“I am glad that the Scottish Government are increasing their side of testing but the majority of testing in Scotland is run by UK bodies, and we are expanding that. There will be another walk- through centre opening in Glasgow in the next few days, so that availability will be there.
“At this point we just ought to say a huge thank-you to the people at the Lighthouse lab in Glasgow who are working 24/7. When they first started off they were doing about 40 tests a day, they’re now doing tens of thousands.”
The new walk-in testing centre at the Arc Sport Centre in Glasgow is due to open on Friday afternoon, the Scotland Office said.
With a walk-in centre already open in St Andrews, Ms Sturgeon said plans are “on track” for a further four such facilities to open in the next two weeks as part of a move to “establish 20 more walk-in centres across Scotland over the autumn and winter”.