The six-time Serie A winning manager would consider taking on the job at the London Stadium at the end of the season
This means the Hammers hierarchy would be open to hiring a caretaker manager to take them to the end of the campaign should Lopetegui be sacked.
Graham Potter has also been linked with replacing the former Wolves boss having also been sounded out by Lopetegui's old club.
The Spaniard was only appointed in the summer, replacing David Moyes in the dugout.
However, he has since won just four of his 14 Premier League matches in charge.
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Following their loss to Leicester in midweek, the Hammers are 14th in the table after one win in their last five.
Lopetegui's side could also find themselves just three points above the bottom three going into the visit of Wolves on Monday.
The 58-year-old was asked about his job security during his pre-match press conference on Friday.
He told reporters: "I understand all the things around football and I understand that my main responsibility is to be ready to face a tough match.
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"It's a big opportunity and challenge for us at home. To be ready to achieve the three points, that is the most important thing."
Meanwhile, Allegri is currently without a club following his second spell in charge of Juventus.
The Italian left the Old Lady in May after three years in charge, having finished third in Serie A and won the Coppa Italia last term.
During his first tenure with Juve, Allegri won five Serie A titles in a row as well as four Coppa Italia trophies.
He also won another league title with AC Milan in 2011 before taking charge in Turin.
A move to the London Stadium would represent Allegri's first managerial role outside of Italy.
The 57-year-old previously revealed back in 2019, though, that he was taking English lessons ahead of a possible Premier League appointment.