The source did not provide any detail about the potential price of the deal though local newspaper Expansion said the building could be worth between 230 million ($246.7 million) and 250 million euros.
The process to start receiving non-binding offers would start on Thursday. "The aim is to finalise a deal around August," the source said.
Caixabank declined to comment, while CBRE was not immediately available.
The sale follows the lender's acquisition of Bankia, which was based in a bigger and newer building in the north of Madrid on La Castellana avenue that has become a landmark.
($1 = 0.9324 euros)
(Reporting by Jesús Aguado; editing by Inti Landauro and Jason Neely)