Solskjaer has endured a hard start to the season, and United are already 15 points off rivals Liverpool at the top after just eight games.
Allegri's potential appointment would also speed up the arrival of Mario Mandzukic in January, while he could also help lure Emre Can back to England.
Griezmann ironise sur le cas Giroud — France-Turquie
Pas forcément mais je ne peux pas me contenter de ce que j'ai aujourd'hui à Chelsea . Je reviens à Chelsea gonflé à bloc, cela fait du bien d'être décisif.
Last season, Jose Mourinho was sacked by the Red Devils before Christmas and the pressure is now building on Solskjaer.
The Italian boss - now out of a job after stepping down as Juventus boss at the end of last season - has promised the former French left back he'll include him among his backroom team in the event of an agreement with United.
With Tuttosport's report and Allegri's latest comments, the reports linking him to United will only increase if the English club suffer an embarrassing defeat at the hands of Liverpool.
U.S. seeks Turkish ceasefire from meeting with Erdogan: Mike Pompeo
But Mr Erdogan has remained unfazed by the pressure, telling reporters: "They tell us "to declare a ceasefire". However, recent developments suggest that Washington may indeed hurt the Turkish presidency.
Tottenham, meanwhile, are also keeping tabs on the ex-Juventus boss with the future of Mauricio Pochettino still unclear.
The situation is expected to be cleared up relatively quickly, with Allegri pushing for United to decide quickly as he wants as much time as possible to revive the club's "already compromised" season.
Loss to Canada speaks volumes about the current state of the USMNT
I think Alphonso is just one part of this unbelievable team. 'You could see the disappointment in his face'. Whilst Arfield didn't score, he set Davies up for the opener and was in inspired form throughout the match.