The Italian giant has endured a tough start to the season, winning only two of its first seven games in Serie A and losing its opening two Champions League group-stage matches for the first time.
And Sissoko – a Juventus player between 2007 and 2011 – feels the club may have to dismiss Allegri if they cannot beat Bologna at home on Sunday in their first match after the international break.
"There's a big problem there. It's true, the start was very bad," Sissoko said.
"We will see what happens after the international break because I think it's going to be a big moment if the team don't win."
Allegri is in his second stint with the club having presided over a hugely successful tenure between 2014 and 2019, winning five straight Scudetti and reaching two Champions League finals.
However, his second reign has been underwhelming. A fourth-place finish last season followed by a slow start this term has led to many fans calling for the Italian coach to be sacked.
And Sissoko believes this feeling around the club is not beneficial for the players, saying: "When you work in this atmosphere it is not easy for players.
"If the result don't come, I think it's better to change, to have fresh management, new people and new things."
Juventus fans will be hoping star signings Angel Di Maria and Paul Pogba can help Juventus climb the table.
Di Maria, signed from PSG in pre-season, has scored once in his first four league games while Pogba is yet to play because of a torn meniscus.
But Sissoko still believes the two signings will be a key part of Juventus' campaign.
"The team need to work a lot – it's a team performance, but if you put more quality in this team, like Pogba, like Di Maria, 100 per cent it's not the same team," he said.
Pogba has also had to contend with off-field controversy in recent months.
His brother, Mathias, has been charged in connection with an alleged armed extortion attempt against the World Cup winner.
Asked how this might affect Pogba’s performances for Juventus when he returns from injury, Sissoko said: "To be honest, it's not easy.
"I think Pogba has the good mentality to support all these things. I think he's going to fix all these things with him and his family to come back to play football and to give a lot of things to the supporters and for himself also."