Eddie Howe is hoping that the Italian can help the Cherries avoid an instant return to the Championship.
The proposed deal would see the 23-year-old move to Dean Court until the end of the season with a view to a potential £10m permanent deal in the summer (via Mail).
While the Italy international was once deemed as one of the brightest talents in Europe just a few years ago, his career - much like his club - has not lived up to expectations in recent years.
With Milan struggling financially (via Mail), El Shaarawy is currently out on loan with Ligue 1 side Monaco, but has started just seven of his 15 appearances for the principality outfit and has failed to score or register an assist.
And due to those financial problems, the Rossoneri are reluctantly willing to let the former Genoa man leave San Siro on a permanent basis in the near future (via Mail). However, the Mail add that Roma are also interested in a move, and are favourites for El Shaarawy's signature due to his desire to remain in his home country.
Despite his recent struggles, there is little question that El Shaarawy is the kind of signing that will capture the imagination of the south coast club, and his talent would certainly aid their attempt to stay in the top flight for another season.
And with the reported offer meaning that the youngster would join Eddie Howe's side on an initial loan with a view to a permanent deal, there will be less pressure on El Shaarawy to hit the ground running, and that may allow the playmaker to show off the attributes that led to him being so highly-rated not so long ago.