Vasseur has been drafted into Maranello for 2023 after his predecessor Mattia Binotto resigned last year amid a belief he did not have the confidence of Ferrari's senior road car management.
And while Vasseur is confident that Ferrari has everything in its hands to be able to fight for the F1 championship this year, Montoya believes there are still elements that the Frenchman will struggle with.
In particular, he thinks the internal political situation at Ferrari is always a difficult one to deal with, thanks to the squad having quite a high turnover of senior personnel when things do not go right.
Speaking to Motorsport.com, Montoya said: "One of the hard things with Ferrari is that I don't think anybody feels like they have job security.
"Everybody is waiting for you to put your foot in the wrong place, so they can fire you. That's what I see from the outside.
"You look at Toto [Wolff] and you know Toto is not going anywhere. You look at Christian [Horner] or you look at Helmut [Marko], they are not going anywhere. But you always look at the head of Ferrari and you are wondering: 'Okay, how long is this one going to last?' Because there is always somebody after him."
Montoya expected Ferrari to have looked to its former boss Jean Todt as Binotto's replacement, and thinks he would have risen above any concerns about losing his job.
"I thought they were going to put Jean Todt in that position," he said. "To be honest, that would have been my pick.
"I don't know if he wanted the job, but he's the guy that did the job before. And I think everybody respects him enough that nobody is going to fuck with him.
"He is a guy that doesn't need the job, so when you don't need the job and you don't care [about losing it], I think you have more balls to make the right decisions.
"But I hope they do a good job. It was kind of surprising, it was such a strong car they had in the beginning of last year, but they didn't use it well.
"And, for the surprise of everybody, you heard them say 'oh, we can still win the next ten races.' And they didn't win a single race."
While Ferrari was Red Bull's closest challenger in 2022, Montoya thinks the situation could be different in the forthcoming campaign.
He reckons that Mercedes, which ended the year strongly having got on top of the issues with its W13, will carry its momentum into the new season.
"If I was Red Bull, I would be looking more to what Mercedes can bring than what Ferrari can bring," added Montoya.
"If you look at the Ferrari pitstops, the strategies, you go like 'who the hell is making those calls?' It's hard to say, because I know under pressure it's tough to make decisions, but if you're not the person making the right decisions, you need to get somebody else, I think. No?
"Or maybe if he is the right person, he needs to have better support around him to be able to do what he does. But something's missing because the mistakes are too often with the strategies they make."
Ultimately, Montoya believes that errors like Ferrari make are triggered by Red Bull being so strong in the way the team operates.
"Red Bull does such a good job with strategies, they make everybody else look stupid," he said. "It's not like anybody else doesn't know what they are doing, but Red Bull has such a better understanding of the strategies and what you need and how to do it.
"Even if they don't make the right call, if Red Bull makes a wrong call, they make it look right. And when somebody else makes the right call, Red Bull makes a better one. So it's like: how do you beat this?"
