The 58-year-old is among the names being considered as United assess their options, with Carrick's future remaining uncertain despite Tuesday's 3-2 Champions League comeback victory against Roma.
According to French transfer journalist Santi Aouna, Manchester United have approached Deschamps, who left his position as France manager after 14 years in charge following the World Cup. He guided his country to the 2018 World Cup title and also reached the final in 2022.
Deschamps has considerable club experience, having won Ligue 1 with Marseille, secured promotion to Serie A with Juventus following their relegation to the second tier and taken Monaco to the 2004 Champions League final, where they lost to Porto.
He is reportedly in no rush to return to management and is not particularly enthusiastic about the prospect of taking charge at Old Trafford, although he is prepared to listen to offers.
Carrick's position has come under pressure after a difficult run of results. United remain 17th in the Premier League, outside the relegation places only on goal difference, and had gone ten matches without a win in all competitions before beating Roma.
That victory offered some relief, with United recovering from two goals down to secure a significant European result. However, the performance did little to remove concerns about the team's consistency or the direction of the project.
There remains support for Carrick within the club hierarchy, and United are understood to be reluctant to make another short-term appointment. The former midfielder was handed the permanent job in May after an impressive spell as interim head coach, during which he helped secure Champions League qualification.
United have also considered other possible candidates, including Julian Nagelsmann, Eddie Howe and Brighton's Fabian Hürzeler. Aston Villa manager Unai Emery has been admired, but the prospect of persuading Villa to release him is considered difficult, particularly with the club keen to secure his long-term future.
Antonio Conte has also been discussed, although United are reportedly unconvinced about the available options. Howe has been linked with the vacancy, while recruitment director Christopher Vivell is understood to have been influential in the consideration of Nagelsmann.
The club are not believed to have made formal approaches to those other candidates, with the preference still being for Carrick to improve results and remain in charge.
Sunday's Premier League meeting with Aston Villa could prove important. The international break that follows will give United's football leadership, led by Jason Wilcox and Omar Berrada, an opportunity to reassess the situation.
Deschamps' reported availability provides another name for United to consider, but there is no indication that a change of manager is imminent. For now, Carrick's immediate task is to build on the Roma victory and produce the results needed to keep his position.