The museum, located within a municipal park in the city's cultural district, is seen by many critics as Kahn's defining work.
The Kimbell Art Foundation commissioned the museum in 1966 after Kay Kimbell, a wealthy industrialist and art collector who died in 1964, mandated the creation of a Texan art institute in his will.
Timothy Potts, the museum's director, has proposed that the new structure be situated on a 2.3ha plot across the street from the original building.
The Kimbell board has yet to reach a decision on the expansion, but it is likely to favour the new construction, which will ease space restrictions.
The museum is close to Tadao Ando's Fort Worth Modern Art Museum, the subject of an international design competition in 1997.
The method by which an architect will be picked for the latest addition to the prestigious artistic quarter has yet to be revealed, but the high-profile project offers a major opportunity for the chosen practice.