The Scottish National Party accused Mr Johnson of “extraordinary arrogance” and said Conservatives’ position on independence was “undemocratic, unsustainable and crumbling”.
Under the terms of Section 30 of the Scotland Act, the prime minister can devolve the power to call a referendum to the Scottish Parliament, as happened in 2014 when Scots voted to stay in the UK.
Asked on Monday whether he would ever take this step as PM, Mr Johnson said: “I’m ruling it out. I think it’s a bad idea.”
But speaking on BBC Radio Scotland, Mr Jack indicated that an outright victory for the SNP in the Scottish Parliament elections of May 2021 would pave the way for a fresh vote.
Asked if winning most of Scotland’s 59 seats on 12 December would give the SNP a democratic mandate for a Section 30 order, the Scottish Secretary replied: “No.”
But asked why not, he replied: “Because the democratic mandate for a Section 30 order is a matter for 2021. We’ll see whether or not the Scottish National Party get a majority then.
“I mean the Scottish National Party - not in collaboration with other parties, not in any alliances - but a Scottish National Party majority.”
Scottish First Minister Nicola Sturgeon has said she wants “IndyRef2” in the second half of 2020, and has made clear that the offer of a fresh referendum will be a condition for SNP support to prop up any minority administration in a hung parliament at Westminster.
Mr Johnson has repeatedly accused Labour of preparing to offer the SNP a referendum in 2020 in return for their support, but Jeremy Corbyn insists he is ruling one out before the 2021 Scottish elections.
The SNP is currently three short of the 65 seats required for an outright majority at Holyrood. Its seizure of overall power with 69 seats in the 2011 election paved the way for the first independence referendum.
Responding to Mr Jack’s comments, the SNP candidate for Dumfries and Galloway, Richard Arkless, said: “It is extraordinary arrogance for Boris Johnson or any other Westminster politician to try to block Scotland's democratic right to choose its own future.
“Whether and when to hold an independence referendum must be a matter for the people of Scotland, not out-of-touch Westminster politicians.
"It is totally unacceptable for Boris Johnson to say he will overrule the people of Scotland if the SNP wins this or any other election, or for Alister Jack now to say the Tories will decide which election does or does not count as a mandate.
"Both Labour and the Tories are all over the place on this issue because their position is undemocratic, unsustainable and crumbling. A vote for the SNP will send a strong message that the future of Scotland is for the people of Scotland to decide not arrogant Westminster Tories."