Bennett, 80, had to postpone Thursday's appearance in Columbus with the Columbus Symphony Orchestra and Friday's show in Cleveland because of congestion and other symptoms.
Doctors suggested that the 80-year-old crooner restrict plane travel and rest for the next five days, according to a press release issued by Cuyahoga Community College, where the singer was scheduled to appear.
Bennett, who recently mentored contestants on "American Idol," is fighting the flu and experiencing congestion.
The flu prompted Bennett to cancel a guest performance on TV's "Idol" earlier this month.
The college said a new date for Bennett's concert will be announced as soon as possible and that the tickets will be honored at the new venue.
"The 15-time Grammy winner was advised by his doctor to restrict plane travel and to rest for the next five days. Both concert appearances are in the process of being rescheduled," publicist Sylvia Weiner said in a statement.
"His condition is improving and he is expected to return to his full concert schedule," which includes U.S. and British appearances later this month and in May.
Bennett, best known for his rendition of "I Left My Heart in San Francisco," lives in New York.