According to Bleacher Report UK, the current Italy manager wants his new club to enquire about the availability of the full-back.
Some recent reports have suggested that Jose Mourinho is not keen on retaining Darmian and could offload him as he goes about cleaning out the Red Devils squad he has inherited from Louis van Gaal.
This has alerted Conte who would like the 26-year-old former Torino star to follow him to Stamford Bridge after this summer's Euro2016 tournament in France.
However, it is unlikely that Mourinho will sanction a transfer between his new club and former employer.
Darmian started on the left side of a five-man midfield as the Azzurri defeated Belgium 2-0 on Monday night as Conte went with a 3-5-2 for their opening Group E match.