Since being launched on Friday, the petition has so far amassed 5000 signatures, with thousands adding their names in the last 24 hours.
Organiser Dean Halliday said: “There is about to be a massive second wave of this new virus in England due to Tory ineptitude.
“The borders of both Scotland and Wales should be closed to all but essential traffic ie commerce.
“Not casual trips that are in fact international from a COVID hot spot about to get worse.
“This is at this time imperative to save thousands and thousands of lives.”
There has been recent discussion over the possibility of introducing a quarantine to anyone visiting Scotland from England if Covid-19 cases rise, with Nicola Sturgeon refusing to rule out the possibility when asked at the daily coronavirus briefing.
The First Minister warned that England does not appear to be trying to eradicate coronavirus.
“I’ve no plans to introduce anything like this just now but I’m not ruling anything out, if it’s required from a public health perspective,” she told reporters on Monday.
“We are taking the time to consider the UK Government’s proposals around air bridges for the arrangements between parts of the UK and other countries.
“If we did see an ongoing divergence between infection rates and levels in Scotland and other parts of the UK, from a public health perspective, we would require to give consideration about how we mitigate that and guard against infection rates rising in Scotland as a result.”
The Prime Minister’s official spokesman said: “On the issue of quarantine we have taken a four-nation approach to these matters throughout. We have worked closely with the devolved administrations at all times and this continues.”