The west London club are now fifth in the table and eight points behind fourth-placed Spurs with just seven games left to play, meaning that Champions League football for next season is looking unlikely.
Allegri, who is already learning English, had talks with Arsenal about replacing Arsene Wenger last summer, but now looks likely to move to west London instead.
Chelsea owner Roman Abramovich is believed to be unhappy that the club look set to miss out on Champions League qualification but is reluctant to sack manager Conte.
Chelsea's top target is thought to be ex-Barcelona boss Luis Enrique, however, Allegri potentially eyeing a move away from Italy would seemingly persuade the Blues to look for other alternatives in the summer.
The Blues are now eight points adrift of Spurs in the final Champions League spot, while Man City can win the title next week, but Chelsea expect Conte to see out the remainder of the campaign with an FA Cup still to play for.
Ethan Couch: 'Affluenza' teen released from jail
Couch's attorneys, Scott Brown and Reagan Wynn, issued a statement Monday saying Couch has always been sorry for what he did. Now that Couch has served his time, he'll have the chance to complete his probation.
The Conner Family is still relevant in America
Trending: Hulk Hogan Returning to WWE? "I think you accused her of being a murderer, didn't you?" asked Kimmel. Barr responded: "I'm still the same". "You all went so far out you lost everybody".
Schools shut in Oklahoma amid teacher protest
He thought it was likely a couple of men who sat in the back of the meeting and called themselves "union supporters". Most of the Oklahoma teachers walked out of classrooms across the state to attend the rally in Oklahoma City.
Michigan, Loyola in Final Four; bluebloods next
Florida State: The loss ended an incredible run for coach Leonard Hamilton's Seminoles, who hit their stride at the flawless time. Still, Hamilton probably isn't the first coach to get testy immediately after losing a regional final in the NCAA Tournament.
Yankees' Home Opener Postponed by Snow
Toronto will close out its seven-game homestand with a three-game series against the Chicago White Sox starting Monday night. Devon Travis added an RBI single for the Blue Jays , who salvaged the split of the series by winning the last two games.