A consulting firm with ties to Bill and Hillary Clinton is eyeing the possibility of an initial public offering, according to people familiar with the matter.

Teneo Holdings, whose clients include chief executives of some of the world’s biggest companies, has benefited from its founders’ association with the Clintons. Douglas Band is a former top aide to Mr. Clinton who helped shape his postpresidential career and created the Clinton Global Initiative. Declan Kelly is a businessman whom Mrs. Clinton appointed U.S. Economic...