Boris Johnson slated Ms Sturgeon during a private meeting of the 1922 Tory backbench committee on Tuesday evening. It is understood he said the vaccine effort was a “UK effort” and had been delivered thanks to the “strength of the Union.”
It came as First Minister Nicola Sturgeon said she hopes to accelerate the vaccine rollout programme if supplies of the jab allow it.
Mr Johnson’s comments came in response to a question which was asked by a Scottish Conservative MP.
One MP, who attended the meeting, told The Guardian: “Essentially, the point the Prime Minister was making is that the UK is a major country, we’ve got sufficient clout to get the vaccines rolled out.
“He did actually mention that we were ahead of the rest of Europe.
“He said if it were up to the SNP then there wouldn’t have been a single vaccine delivered in Scotland.
“It was a UK effort, in other words, and needed the clout of a big government.”
A spokesperson for Mr Johnson, added: “There is no doubt that the Union has been critical in the development, production and administration of the vaccines and indeed across a range of measures during this pandemic we have all worked together to provide for the British people.”
Ms Sturgeon said during her daily coronavirus briefing everyone over the age of 50 and younger people with underlying health conditions should receive at least the first dose of either Oxford/AstraZeneca or Pfizer vaccine by May.
“Clearly, I’m not going to stand here and do that. But the 12 weeks is the advice that we will be designing our vaccination programme around.”
More than 100,000 people have received their first dose of the Pfizer vaccine but ministers expect to have access to more than 900,000 doses of both vaccines, roughly split evenly between the two, by the end of January.
The UK has bought 100 million doses of Oxford vaccine and Scotland will get a proportionate 8.2 per cent, which has been paid for by the Treasury.
Doses will be given first to care home residents and their carers, people over the age of 80 and frontline health and social care workers.
The programme will then be extended to the rest of the population, starting with people aged 75 to 79, followed next by those aged 70-74 and the clinically extremely vulnerable.
It comes as a total of 146,024 people had tested positive in Scotland since the start of the pandemic.
Of the patients in hospital, 102 are in intensive care – up by two on Thursday.
New cases included 692 in the Greater Glasgow and Clyde NHS area, 390 in Lanarkshire, 220 in Lothian and 192 in Grampian.