Deloitte LLP has told partners Chief Executive Cathy Engelbert hasn’t been renominated for a new term, touching off a behind-the-scenes leadership dispute at the Big Four accounting firm.

Ms. Engelbert, the first woman to run one of the Big Four firms in the U.S., is three-plus years into a four-year term as Deloitte’s CEO. But Deloitte’s U.S. partners were told in a recent email that she wasn’t being renominated for a second term at this time as the firm begins its every-four-years CEO-election process, according to people...