In a letter sent to Silver on Monday, the Globetrotters called on the NBA to reward the team's work in helping to spread basketball worldwide by welcoming it into the league.
"As an organization whose storied history is already tightly interwoven within that of the NBA, the Harlem Globetrotters are looking for a long overdue seat at the table," Globetrotters general manager Jeff Muun said in a statement.
"Our players were instrumental in the integration of the league dating back to 1949. We stood proudly as our players were recruited by NBA teams."
"Now after years of attracting the best Black players, it's time to the NBA recognized our contribution to the game.
"With the league already considering an expansion, the time has come. The Harlem Globetrotters stand ready to negotiate for a franchise."
Silver has fuelled speculation that the NBA is considering adding a 31st team to the league, telling ESPN in an interview last December that expansion was "inevitable."
"It's sort of the manifest destiny of the league that you expand at some point," Silver said at the time.
Seattle has long been considered the front runner for any new NBA franchise. The city has been without a team since the SuperSonics moved to Oklahoma City in 2008, and would in theory have a stronger case than the Globetrotters given that New York already has two NBA franchises.
The Globetrotters were founded in 1926, and are best known as a traveling exhibition team having played thousands of games in 122 countries around the world.
However the team has contributed several key moments in NBA history, most notably when Globetrotter Nathaniel Clifton became the first Black player to be signed by the league in 1950.
The same year, the Globetrotters' Chuck Cooper became the first Black player to be drafted when he was chosen by the Boston Celtics with the 14th pick overall.