Giroud scored all four goals in Chelsea’s 4-0 win over Sevilla on Wednesday night and became the oldest player to score a Champions League hat-trick.
But that performance has not ended the speculation over his future at Chelsea, where he has been aching to play more football this season.
Tammy Abraham is preferred over the Frenchman at Chelsea for the moment and the striker is believed to be prepared to leave the club if he is not guaranteed more playing time in the second half of the season.
Inter are still interested in the veteran striker and, according to Italian daily Tuttosport, he is the preferred option for Conte in the January transfer window.
The Serie A giants also hold an interest in Napoli hitman Milik who is also expected to find a new club next month.
But Conte would prefer to sign Giroud, with whom he worked well when he was the Chelsea manager.
The Frenchman came close to joining Inter in January last year before the Nerazzurri backed out of the negotiations.
But the Italians are keen to sign him this time around and Conte wants to reunite with his former Chelsea striker at the San Siro.