El Shaarawy moved to Milan from Genoa last summer but has found it hard to displace the team's more well known names like Zlatan Ibrahimovic, Robinho and Alexandre Pato from the first-team squad.
The Rossoneri also appear set to sign Manchester City striker Carlos Tevez which will realistically push El Shaarawy further down the pecking order.
Sky Sport Italia claims that the German Bundesliga outfit want to bring the player on-loan for at least six months.
It reports that Cologne hope El Shaarawy's arrival will ease the burden on star striker Lukas Podolski for goals.
The 19-year-old had previously linked with a move to Fiorentina in the winter transfer market.