Unzue, who was assistant to Luis Enrique between 2014 and 2017 at the Camp Nou, said the standards demanded by the Argentine are too much for some teammates.
“What has kept Messi [where he is] is ambition and how to live with moments of stress,” he told YouTube channel Idolos.
“He self-demands to himself to keep being the best and, as a result, also wants the best around him, teammates, physio, or coach. That produces a great stress that some teammates are not capable of bearing.”
Messi almost left Barca in the last transfer window amid speculation of a move to Manchester City and many expect him to leave the club when his contract expires in June.
The 53-year-old Unzue said the situation surprised him.
“The truth is it came as a shock to me that Messi wanted to leave but, knowing the club and Leo, I understand that he might have very, very important reasons for taking this decision.
“What I was convinced of was that if, in the end, he stayed, he’d give his best and maintain his ambitions.
“[Barcelona coach Ronald] Koeman is managing Messi and the group well, and the situation is not easy, although I think entering a big club that has won titles is harder when they’ve stopped winning.
“When we arrived with Luis Enrique we did so after a year the team had not won anything.
“That’s happening to Koeman now, he’s arriving at a messed-up time but if it’s not a messed-up time, you don’t get the call to become coach of Barcelona.”