Conte's Inter finished the Serie A season with a 2-0 win over Atalanta, ending the campaign just a point behind champions Juve.
The standings fail to tell the full story, however: their title chances had realistically collapsed before lockdown, and they sat seven points behind Juve a week ago, at the point Maurizio Sarri's team secured the league for a ninth straight season.
Since arriving in the summer, Conte has spent more than £170m on new signings, but has nonetheless been openly critical of the Inter board. CEO Beppe Marotta was forced to publicly distance himself from Conte's comments in the wake of the Atalanta win, which took aim at president Steven Zhang, claiming that he had been given 'no protection' from those above him.
These claims look to have crossed a line with the Inter hierarchy, and Sky Italia report that unless something 'changes sensationally', Inter will move in another direction this summer.
Allegri, who has been out of work for more than a year, is said to be in 'pole position', as Zhang looks for a new manager to implement his vision for the club.
Speaking to Sky Italia following the Atalanta win, Conte said: "Neither my work nor the players' work was acknowledged, I found little protection from the club when making comparisons. If you want to reduce the gap with Juventus you have to be strong on the pitch but above all off it.
"We will have to talk to the president who is in China now. I don't like people jumping on the bandwagon now, we have to be there for the good times and bad and this has not happened at Inter. I understand that, if you are weak, it is hard to protect your team and coach."
Pretty remarkable stuff from a guy who's signed Romelu Lukaku, Christian Eriksen, Nicolo Barella, Diego Godin and Alexis Sanchez all in the last year, but each to their own.