I don't remember if this has been discussed. Did Barzini contact Tessio before Vito had died?
If so, Tessio would not only betray Michael, but also Vito, which I can't believe.
If not: There wasn't much time between Vito's death and his funeral. And Barzini must have calculated some days for Tessio to grieve for his boss and friend, before he opened up for his suggestion.

I think Tessio would have had feelers out since at least that meeting that Clemenza and he had had with Vito and Mike.

I agree with Lilo. In the fishtank scene, Vito asks, "Do you trust my judgment?" "Always, Godfather," says Clem, with feeling. "Yess-ss," hisses Tess. If I remember correctly, Tess exits hastily, without a handshake. I think he sent out feelers to Barzini soon afterward. I don't think Barz attempted to turn him first, much less at Vito's burial.

I thought it was Tessio who made the handshake with Michael.

Tess shakes Michael's hand, but does so with his LEFT hand. This is telling. Much like when Luca refused to shake hands with Tatt Jr.

I am sure Tess put feelers out right after the meeting, and it was even clear to an ailing Vito that there would be a traitor in the family as soon as he died.

Tess and Barz, who underestimated Michael, figured he'd be easy game once he took over.

It's at least ironic that Tessio ("Tessio was always smarter.") was the one who "... sensed something else in the man[Michael]; a force cleverly kept hidden, a man jealously guarding his true strength from public gaze ..."

Tess did not know about the secret Neri Regime, so he thought it didn't matter how strong Michael was because he figured Barz and the other families would have more muscle than Clemenza.

...plus, I think Barzini promised Tess that he'd be the undisputed head of what remained of the Corleone family.

In the fishtank scene, Clemenza says, "Don Corleone, you said the day would come when me an' Tessio could form our own family" [emphasis added]. One family with co-Dons? Never happen. Tess and Clem either would have had to divvy up the Corleone Family, or fought to the death to be the undisputed Don. When Tess went over to Barzini, I'm sure that part of the deal was that Clem would be whacked along with Michael.

The irony is, though, even had Tessio and Barzini succeeded, Tessio's days surely would have been numbered. If Tessio was proven to be disloyal to the Corleones, how could Barzini have expected any loyalty at all from Tessio either? Someone as shrewd as Barzini would see Tessio's behavior as a deep character flaw, one that may be temporarily quelled but eventually would manifest again under certain circumstances.

In all likelihood, once The Corleones and all internal threats from within the family had been extinguished, Barzini would have had Tessio eliminated as well, replacing him with a figurehead whose loyalty had been proven and unquestioned.

I agree with Blonde. That's the way the mafia works. Funzi Tieri also encoured members from Angelo Bruno's family to turn against their boss. But after Bruno was killed Funzi ordered the death of the same men he had encoured to kill Bruno, because they had shown to be disloyal. A same thing also happened with Bonanno zip Cesare Bonventre who had conspired with others to kill Carmine Galante, but was eventually wacked himself as he was seen as a liability.

What Barzini did would be close to what Lucky did with Salavtore...after taking out Messeria, he took out Salavtore because he knew Sal would viewed him as disloyal and already planning to have him killed. So it makes me wonder, did Tessio have a backup plan? ....did Tessio have something in place to take out Barzini soon as possible (had Vito actually been killed)?

Matter of fact, I believe that it was quite the opposite. Barzini was the one who probably had a back up plan to take out Tessio had they succeeded in assassinating Michael. Barzini's thinking:

"Now how can I ever trust a man who is willing to set up and sell out the Corleone family...a family that he has been with since the beginning? If he was willing to turn on Vito and his son, a family that has been loyal to him, then who am I? Surely he won't think twice about turning on me if the opportunity presents itself."

I believe that Barzini made all kinds of promises to Tessio and never intended on keeping them. He was going to take Tessio out as soon as he, Barzini, became the new Boss of Bosses.

Although Mike considered Tessio smarter than Clemenza, that might not be saying much. In any case, Tessio was an operations manager, not an executive. His concern was survival; he wasn't a strategic thinker.

I agree with both. Clearly Barzini would know that a guy who turned on his long-term Godfather would be an immediate security risk to his new boss or patron. And, having gotten rid of Michael (and presumably Clemenza), why would Barzini tolerate even a semblance of independence from Tessio? So I agree that he would have dispatched Tess sooner rather than later.

But wouldn't Tess, "always smarter," have figured that out? Sure. But, a constant theme in the novel and the trilogy is how greed clouds men's judgment. Probably Tatt believed all the blandishments Barzini offered him because he wanted to believe them, rather than think about how much risk he was putting himself to. So he ignored the peril--just as Paulie Gatto and Carlo Rizzi did.

I remember I already said this in another topic. But a scenario such like this happened in real life. Cesare Bonventre, Carmine Galante's bodyguard, conpsired with Galante's enemies to take him out. After that, he was promised to become a caporegime of the crew of zips. He did become a caporegime, but was killed a few years later anyway. I'm not sure about the exact reason for this, but I remember reading/hearing that it had something to do with him betraying Galante. He was therefore considered unreliable.

Bonventre's assassination was part of Joe Massino's long-term plan to eliminate every rival. Galante was the champion of the Sicilian faction. At one point, another Zip, Sal Catalano, was effectively the street boss of the Bonannos. Bonventre switched sides. But he was still a Zip as far as Massino was concerned--not an American, not his guy (like Vitale)--a potential rival.

How come Massino didn't eliminate Sal Catalano?

Well I wouldn't give Mike too much credit. Vito basically told him what would happen. Vito told him that whoever approached him to set a up a meeting would be a traitor and Tessio was the one who did so.