Mourinho has come under fire and was reportedly on the brink of the sack before he masterminded a stunning comeback against Newcastle last weekend.
But Calciomercato claim the club are keeping their options open, with Juve chief Allegri their first-choice if they decide to wield their axe at Mourinho.
They say United chiefs have already made contact with the Italian’s agent and were told that Allegri would be interested in taking the reins at Old Trafford.
Club officials have previously insisted Mourinho’s job is safe but that could change after the international break.
The 20-time Premier League champions have a tough run of fixtures when their players return from their countries.
They travel to Chelsea in their next outing on October 20, before two games with Allegri’s Juve in the Champions League (Oct 23 (h) and Nov 7(a))
Their November fixtures also include trips to high-flying Bournemouth (Nov 3) and local rivals Manchester City (Nov 11).
And Mourinho, 55, could be back under the spotlight if results do not go his way,
United are currently languishing in eighth in the Premier League table, five points off the Champions League places and another two off leaders Manchester City.
Meanwhile, Allegri, 51, has guided Juve to their best-ever start to a season. The reigning Italian champions have won al their ten games in all competitions in 2017-18.
He has won four Serie A titles during his time in in Turin, four Coppa Italia titles and is a two-time Champions League runner-up.
Allegri also guided AC Milan to there last Scudetto in 2010-11.
He has been linked with a move to England in the past, with Arsenal thought to have considered him as a replacement for Arsene Wenger.
His current contract with Juve expires in 2020.