The former European giants are 13th in Serie A -- three points above the relegation zone.
Giampaolo, 52, replaced Gennaro Gattuso during the summer, but the former Sampdoria boss has suffered four defeats in his first seven games, including a 2-0 loss to city rivals Inter.
Milan went on to lose their next two matches against Torino and Fiorentina before last weekend's 2-1 victory at Genoa.
Former Inter Milan coach Luciano Spalletti is the 18-time Italian champions' first choice to replace Giampaolo, according to Gazzetta Dello Sport and Sky Sport Italia.
However, the 60-year-old has not been able to agree severance terms with Inter, who he guided back to the Champions League the past two seasons, and remains under contract until 2021, despite being replaced by Antonio Conte.
Former Inter and Fiorentina coach Stefano Pioli is now reported to be the favourite to take over the former seven-time European champions, who won their last title in Serie A in 2011.
Pioli previously coached Inter during the 2016-17 campaign and has reportedly been in the running to take over at struggling Genoa.
Former Roma and Marseille manager Rudi Garcia has also been touted among the possible candidates.
AC Milan finished fifth last season but lost their Europa League berth after accepting a European ban for breaching financial fair play rules.