The 32-year-old returned to the Cherries last week after spending the first half of the season on loan at FK Qarabag in Azerbaijan.
Begovic, a £10million signing from Chelsea in 2017, started the campaign as Bournemouth's third-choice goalkeeper behind Aaron Ramsdale and Artur Boruc.
He lost his starting place after a string of errors last term.
And now, Begovic is undergoing a medical at AC Milan to replace the club’s No.2 Pepe Reina, who is close to joining relegation-threatened Aston Villa on loan.
Begovic will provide back-up to Milan and Italy goalkeeper Gianluigi Donnarumma.
Last week, Reina's agent, Manuel Garcia Quilon, confirmed his client would only be allowed to leave if a replacement was signed first.
He said: "There's no agreement. We continue to negotiate, but the Rossoneri would like to bring in a replacement first."
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Aston Villa have a goalkeeping crisis on their hands with long-term injuries to Tom Heaton (knee) and Jed Steer (calf), with third-choice stopper Orjan Nyland starting the last four games.
Reina was seen in the stands for Villa's home game against Man City on Sunday.
The 37-year-old played 394 games for Liverpool in a nine-year spell from 2005 and 2014.
Bournemouth goalkeeper Asmir Begovic is set to join AC Milan on loan.
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