The 20-year-old scored 18 goals with Red Bull Salzburg last season, earning him a move to sister club RB Leipzig over the summer.
However, Sesko has struggled for game time behind Lois Openda and Yussuf Poulsen, starting just four Bundesliga games this season.
With Giroud turning 38 in September and hinting at an exit, Gazzetta dello Sport claim that Milan are looking at Sesko as a younger, like-for-like replacement.
A fee of around €40m (£34m) is being touted for the Slovenian, but that could rise in the summer should the striker have a fine European Championship campaign with his country.
Sesko was the pivotal factor in qualifying Slovenia for their first Euros since 2000, scoring five goals in their section, and he is set to lead the line in Germany in the summer.
Milan will also be looking to bridge the gap between themselves and rivals Inter Milan and Juventus next season, who are currently a long way clear at the top of Serie A.
Meanwhile, Arsenal full-back Nuno Tavares could make a return to Marseille this month, after failing to break into the Nottingham Forest side during his loan spell so far this season.
The 23-year-old spent last season on loan at the Stade Velodrome, scoring six goals from defence in 31 Ligue 1 appearances.
Marseille opted to sign Renan Lodi over the summer instead though, but the Brazilian has since departed to join Al-Hilal in the Saudi Pro League.
The French club have already moved for Young Boys left-back Ulisses Garcia to help plug the gap, but L'Equipe claim that Marseille want another option in that position, and will look to bring Tavares back to the club.
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