The Italian coach has won the Premier League title and has expressed interest in managing at Old Trafford.
According to the Evening Standard, Conte is believed to be keen on replacing Carrick, with reports linking him to the Manchester United managerial role continuing to surface.
After losing to AFC Bournemouth at Old Trafford on Sunday, Manchester United have now gone eight games without a win, leaving Carrick's position precarious.
Manchester United executives Jason Wilcox and Omar Berrada are holding a crisis meeting at the club's training ground today, with the club widely expected to release a statement regarding Carrick's future soon.
Conte left Napoli at the end of last season and is currently available. He has been viewed as a potential successor to Carrick since Manchester United's performance began to decline.
Conte has a successful track record in the Premier League, having led Chelsea to the title in 2017. He has made clear his interest in returning to England for his next coaching role.
In an interview with The Athletic last October, Conte discussed his potential next steps. He stated that he wanted to take over a club that was once great but is now struggling.
Manchester United are undoubtedly England's most successful club, but they have not won the Premier League since 2013, which perfectly matches Conte's description.
Conte said: "If you arrive at a club whose history speaks for itself – a history made of trophies, league titles and victories – you can only respect it in one way: by being ambitious, and when this club is not where it should be, by taking it back and fighting for these goals."
Conte said: "I am well aware that only one team can win every year. But I must instil in my heart, and also inject this spark and hope into the fans, players and club management, to make everyone feel, 'We can win'. Only by winning can you create history."
Conte also issued a statement via his personal social media, directly addressing links with the Manchester United managerial position.
Last week, Conte responded on social media to reports linking him with "several clubs." He emphasized that he had not yet begun negotiations with any suitors and denied reports that he was demanding an exorbitant salary.
However, Conte made clear that he is open to returning to coaching if the right opportunity arises.
The statement read: "In response to recent reports in the Italian and international press, I feel it is necessary to clarify my position."
"My name has been linked to several clubs, but no negotiations have reached an advanced stage at present. After a wonderful and successful time managing Napoli, I will take the necessary time to consider how and when I will return to coaching. This will be a carefully considered decision, made only when I believe the conditions are right."
"I have seen reports stating that I have made financial demands and demands for full control to the aforementioned clubs, and there have also been reports mentioning concerns about my so-called 'tough personality.' These reports are unfounded, and at present, these judgments have no concrete factual basis."
"If a concrete opportunity arises, I will then engage in discussions with the relevant clubs and jointly consider the terms of a potential agreement, with full respect for the needs of both parties and my characteristics and methods as a coach."
"Until then, any speculation about my demands and professional future is mere conjecture and should not be treated as fact."