The Ghanaian midfielder was previously linked with Stoke City.
After taking the Cherries beyond the 40-point mark in their first ever top-flight campaign, manager Eddie Howe is now looking to build for next season and £4 million-rated Agyemang-Badu is reportedly on his shortlist.
The 25-year-old is a regular for Udinese and has scored four goals in 27 Serie A appearances this season.
He has also featured on 62 occasions for Ghana, having made his debut as a 17-year-old.
Stoke City were previously linked with a move for Ageymang-Badu, but the Potters signed Porto's Giannelli Imbula instead.
In January last year, the self-confessed Manchester United supporter told Goal that he would welcome a move to Liverpool following reports in the Italian media citing interest from the Reds.
And moving to Bournemouth may be Agyemang-Badu's best shot at playing for either of the aforementioned pair.
The Cherries moved to 11th following Saturday's win at Aston Villa and are only three points behind reigning champions Chelsea, who occupy the final top-half place.
However, the Blues could extend their advantage by winning their game in hand next month.