Writing in today's The National, Mr McLeish said "of course there will be a second referendum".
SNP leader Nicola Sturgeon this week made a direct pitch to Labour members, whom she said were now considering backing indyref2.
The First Minister set out her mandate for a second referendum at Bute House this morning.
"That's why I'm encouraged by the comments of some senior Labour MPs that the party has to have a rethink about the constitutional question.
"Of course there will be a second referendum. You cannot continue to fight democracy because if you oppose indyref2, you are opposing democracy.
"For Labour, that is not a visions, it's not a strategy, it's not a policy. It's an acceptance of defeatism.
"The only reason they might not want to have a referendum, along with the Conservatives, is that they might lose."
"My best bet is that a referendum will take place after that in either 2022 or 2023.
"I don't believe for a minute that Scotland couldn't be independent, of course it could. But that's not the question we need to be asking ourselves."
Mr McLeish, who briefly served as First Minister from 2000 to 2001, described the SNP as a "competent government", but said not all those who voted for the party at this month's Westminster elections would necessarily be independence supporters.
"There's a danger of looking into the SNP vote and thinking that everybody who votes SNP supports independence," he wrote.
"The SNP have been a competent government, and in many respects a good government, so therefore people vote for them for a variety of reasons.
"LAbour can only get back into Scottish politics if it uses the key of the constitutional question to unlock that political door.
"We're currently nowhere on the Scottish question. I have seen one person suggest that it's nothing to do with Labour, leave it to the Tories and the SNP. That doesn't make sense, it's also quite dangerous.
"Labour has now got to seize the opportunity to device whether it can develop an alternative, or if it can't, what is the next best option."
Mr McLeish said he wanted to see Scottish Labour, led by Richard Leonard, "test the boundaries of all reserved issues", particularly in areas such as welfare and immigration.
"Labour has to rebuild trust," he said.
Mr McLeish served as Minister of State for Scotland from 1997-99 and was leader of the Scottish Labour Party from 2000-01, helping to secure devolution for Scotland in those roles.