The polling expert explained the vote to leave the European Union had driven support up, but that it was no longer the driving factor in support.
Speaking at a UK in a Changing Europe event on Tuesday, Sir John also suggested it was impossible to currently call who would win a second vote on Scottish independence.
He said: “One thing that Brexit has changed is the level and the character of support for independence in Scotland.
“The debate north of the border, leaving aside for the moment the debate in Northern Ireland, is probably going to be the next chapter in the domestic – at least Brexit – debate.
“What evidently happened in 2019 was a step change, that is but by the time we got to the end of that year we were looking at 55 per cent support for independence amongst remain voters, and 50 per cent among leave voters.
“If you then look at what happened during the course of last year, for a while support for independence in Scotland hit the 54 per cent mark on average in the polls. That increase in support occurred pretty much evenly amongst remain and leave voters.
"The gap remained as it was, but the increase was occurring amongst both sides, so the rising support for independence last year was not apparently anything to do with Brexit and seemed to be everything to do with differential perceptions of the way in which the Scottish Government and the UK Government were thought to be handling Brexit and how that seemed to be affecting some people.
“There isn’t any evidence that the Brexit divide at the moment at least is increasing support for independence beyond what it achieved in 2019, if indeed more people in Scotland were to be convinced that Brexit was not a good idea.
“It will depend on the politics of the ability of both sides of the argument to persuade people that Brexit is indeed doing damage or not.”
Sir John also discussed the prospect of another referendum, describing it as an “enormous gamble” for both sides.
He said: “Nobody knows who would win a second independence referendum.
“It’s very clear both sides have a very substantial interest in trying to move the numbers from where they are at present before we get anywhere close to any second independence referendum because at the moment neither side are where they want to be.”
Polling by Savanta ComRes for The Scotsman published mid last month showed support for Yes at 43 per cent if a second independence referendum was held tomorrow, with No on 47 per cent and ‘don’t knows’ at 8 per cent.
Sir John’s comments come as Nicola Sturgeon claimed Downing Street was attempting to “rig” the result of a second referendum on Scottish independence.
Reports on Monday suggested UK ministers were pushing to allow Scots living in the rest of the UK a say in any future votes on Scotland’s place in the Union.