Dreesen said an agreement had been struck "in principle" with the Spanish club, with the 2010 World Cup winner due in Munich later on Thursday for a medical and to discuss terms.
The 32-year-old would become the fifth Spanish player on the Bavarians' books together with Pepe Reina, Juan Bernat, Thiago and Javi Martinez.
Bayern on Wednesday completed the signing of Mehdi Benatia from Roma for a fee of £20.7million and they were not expected to venture into the transfer market again this summer.
Alonso's arrival could see Xherdan Shaqiri seek a move away from the club.
The Switzerland international winger is attracting interest in Italy, with Juventus and Inter Milan both keen on the former Basel midfielder.