Nicola Sturgeon has told people not to travel to or from England except for "essential purposes".
The first minister issued the appeal as Boris Johnson prepared to announce a month-long lockdown south of the border.
Ms Sturgeon said her government would "take account" of the developments but would base decisions on Scottish measures on circumstances in Scotland.
She said there were early signs existing action was having an impact.
Boris Johnson has announced the closure of non-essential shops and hospitality, although schools and universities will remain open.
It followed scientific modelling which suggested the number of deaths in the UK could rise much higher than anticipated, possibly as high as 4,000 a day by mid December, without further action.
Earlier on Saturday, as speculation about new restrictions mounted, Ms Sturgeon indicated that Scotland would continue with its regional tiered strategy for the time being.
A new five-level system of restrictions will come into force in Scotland on Monday, which will see travel restrictions imposed on many Scots.
She tweeted: "Prevalance of the virus is currently lower than in other parts of the UK and there are early signs that the tough restrictions in place since we moved quickly in late September have started to slow the rate of the increase".
4/ Most importantly, we urge everyone to comply with current restrictions, including on travel. People should not travel to or from level 3 areas in Scotland and for now, we are asking people not to travel to or from England at all, except for essential purposes.— Nicola Sturgeon (@NicolaSturgeon) October 31, 2020
She urged people not to travel to or from areas in Scotland under level 3 restrictions and added: "For now, we are asking people not to travel to or from England at all, except for essential purposes."
Ms Sturgeon said decisions on action beyond this would consider what financial support was available as a consequence of any new restrictions in England.
Earlier public health expert Prof Linda Bauld warned the path of the pandemic in Scotland over the next fortnight would determine whether the country goes into lockdown.
She told the BBC's Good Morning Scotland: "You can see from the figures in Scotland that the case numbers are levelling off.
"We are making progress. The question I would ask is the progress quick enough?
On Saturday the Scottish government confirmed a further 1,101 positive cases and 24 deaths.
Prof Bauld said these figures are a "reflection of infection rates in September."
She told the programme significant action has already been taken in Scotland, Wales and Northern Ireland.
Prof Bauld added: "I think the big debate is 'Why is England dragging its feet?' That is essentially the issue."
Prof Bauld, of Edinburgh University, said: "Along with the other devolved nations you can see from the latest ONS data that Scotland is really in quite a different position from many parts of England so I would not assume that we are going to go the same way as England."
But looking ahead, Prof Bauld said: "We need to continue to expect these infection rates to go up and I think that the government strategy in Scotland is to keep an eye on the situation over the next week or two.
"My feeling would be unless we see improvements, particularly in some parts of the central belt, we really, realistically, could be looking at level 4, which we want to avoid but would be necessary, essentially just to avoid these preventable deaths escalating even further."
She also cautioned that in order to get the R-number below one it is possible the existing measures will not be sufficient.
The latest estimate for Scotland is that the R-number is between 1 and 1.3, a slight fall on the previous estimate.
Prof Bauld said: "If the trends go in the wrong direction again then I think everybody in the country recognises that more action will be taken."
'Twisting the rules'
The Scottish government has meanwhile launched a new campaign to warn the general public of the serious implications of "twisting the rules".
It highlights how even small lapses in compliance have an impact not only on health, but the economy and wider society.
Launching the campaign, Ms Sturgeon acknowledged the "huge sacrifices" that have been made since March.
She added: "I know that people are tired and frustrated, but at this critical point in the pandemic, I want to remind people that the decisions they make over the coming days and weeks have a real impact on not only themselves but others.
"Right now, we rely more than ever on public willingness to adhere to the measures in place."
Ms Sturgeon said the new new protection levels should enable communities to "control outbreaks quickly and effectively and minimise transmission of the virus".
But she added: "If we all put our own twist on the rules, they simply won't work."