CalcioMercato.com (h/t Alex Harris of the Star) reported the Red Devils have offered the 51-year-old a three-year deal worth £7.9 million (€9 million) per season.
Mourinho's best chances of survival appear reliant on a good cup run, with the club facing Allegri's team in the next match of the UEFA Champions League group stage. Defeat in that game could cause irreversible damage to Mourinho's standing at the Theatre of Dreams.
Allegri has been a huge success since taking over the Old Lady in 2014, winning four domestic doubles. The former AC Milan boss has nothing left to prove in his homeland, and he could be tempted to follow compatriot and former Napoli boss Maurizio Sarri to England.
Italian coaches have achieved huge recent success in English football, with Carlo Ancelotti, Antonio Conte, Roberto Mancini and Claudio Ranieri becoming Premier League champions in the past 10 years.
However, United fans crave a return to the swashbuckling football of previous years, and it's questionable if Allegri would produce anything close to this ideal. His Juventus side has often made defensive resilience a priority, although they have rarely lacked for goals on their road to dominance in Italy.
Allegri previously managed United superstar Paul Pogba at Juve, which could give him a good chance for success if he takes over. According to Liam Corless of the Manchester Evening News, the Italian complimented the player after his success at the 2018 FIFA World Cup with France.
"I can only say I'm happy for having seen him grow up in an important competition like the World Cup," Allegri said. "He played with great personality. Compared to the past, he matured a lot."
The clock is ticking on Mourinho, and with every defeat or poor performance the incessant pressure continues to build. Neither David Moyes nor Louis van Gaal allowed to remain at the club once it became clear there was no light at the end of the tunnel, and this season's performances have not been good enough.
However, Allegri shows a similar pragmatism to the Portuguese,, and if the United board are happy with such stylistic values, they should stick with Mourinho—who has the added benefit of knowing the Premier League inside out.