Nicola Sturgeon has suggested that Boris Johnson should not visit Scotland as it is not an "essential" journey.
The prime minister is widely expected to travel to Scotland on Thursday.
But Ms Sturgeon said she was "not ecstatic" about the plan, saying leaders should abide by the same rules as they ask of the general public.
Asked about the trip, Scottish Secretary Alister Jack said Mr Johnson would go "wherever he needs to go in his vital work against this pandemic".
And Downing Street has insisted that it is important for the prime minister to be "visible and accessible" during the pandemic.
The prime minister's official spokesman did not confirm details of the visit, but said: "It remains the fact that it is a fundamental role of the PM to be the physical representative of the UK government".
The spokesman added: "It is right that he is visible and accessible to businesses, communities and the public across all parts of the UK, especially during the pandemic."
Reports have suggested Mr Johnson is due to visit Scotland on Thursday to thank staff involved in the fight against Covid-19, despite the "stay at home" lockdown in place across the country.
Speaking at her daily coronavirus briefing, Ms Sturgeon stressed that she was not saying Mr Johnson was unwelcome in Scotland, but added that she was "not ecstatic" about the idea of him travelling up from London.
She said: "We are living in a global pandemic and every day I stand and look down the camera and say 'don't travel unless it is essential, work from home if you possibly can' - that has to apply to all of us.
"People like me and Boris Johnson have to be in work for reasons people understand, but we don't have to travel across the UK. We have a duty to lead by example."
Ms Sturgeon said her team had suggested she visit a mass vaccination centre in Aberdeen in the coming weeks, but that she had questioned whether the journey was "genuinely essential".
She said: "If I'm standing here every day saying to all of you watching, don't leave your house unless it is essential, I have a duty to subject myself to that same discipline and decision making.
"I would say me travelling from Edinburgh to Aberdeen to visit a vaccine centre is not essential - Boris Johnson travelling from London to wherever in Scotland to do the same is not essential.
"If we're asking other people to abide by that then I'm sorry, I think it's incumbent on us to do likewise."
The Scottish secretary was asked about the move at Westminster by SNP MP Neale Hanvey, who described the trip as a "futile" attempt to bolster the union following a trend of polls suggesting majority support for independence.
Mr Jack replied: "That's ridiculous - the prime minister is the prime minister of the United Kingdom, and wherever he needs to go in his vital work against this pandemic, he will go."