Cherries boss Eddie Howe has previously played down his chances of adding to his squad in January, but did admit arrivals could happen if the right player became available.
Musonda, 21, has managed six appearances in all competitions for Chelsea this season and has also featured for the youth team in the Checkatrade Trophy.
However, the recent arrival of Ross Barkley is likely to see his chances limited further in the second half of the season.
Chelsea are keen to send the Belgian starlet out on loan and Bournemouth are reportedly keen after Howe was spotted in the crowd at Chelsea's 2-1 win at Portsmouth in the Checkatrade Trophy on Tuesday.
Bournemouth are likely to face competition although they appear well placed to win the battle for his signature as they can offer Premier League football and have good links with the Stamford Bridge side, having taken Nathan Ake on loan last season before signing the defender outright over the summer.