La Gazzetta dello Sport reports the two Bundesliga sides are interested in signing the Norwegian youngster, who moved to the Rossoneri last year.
Hauge was not called up for the friendly against OGC Nice on Saturday, which fuelled the rumours of a possible move away from San Siro.
The 21-year-old signed a five-year deal with the Diavolo upon his arrival in October 2020 and contributed three goals in five Europa League games, plus two goals in 18 Serie A games with the Rossoneri.
The left winger struggled to establish himself in Stefano Pioli’s plans and Milan are looking elsewhere to bolster their attack ahead of a season in the Champions League.
But the newspaper claims the club is only willing to sell on a permanent deal, if a suitable offer arrives for the former Bodo/Glimt man.
The report believes Milan have set a price-tag at €15m for the former Eliteserien player.