ALBANY — Acting to avert a leadership crisis in New York City’s schools amid a legislative stall in the Capitol, Gov. Andrew M. Cuomo intends to call a special session of the State Legislature as early as Wednesday and introduce a bill that would extend mayoral control of the city’s educational system for one year.

The session would be focused on granting Mayor Bill de Blasio another year of control over the city’s schools and their 1.1 million students, according to an administration official who spoke on the condition of anonymity because of the sensitive nature of the negotiations. The law giving the mayor that power expires after Friday.

Lawmakers left Albany last week with the issue of mayoral control unresolved, as Republicans and Democrats were unable to reach a deal. Mr. Cuomo, a moderate Democrat with a record of working with Republicans, was not able to broker an agreement by the scheduled end of the session on Wednesday.

The looming lapse was deeply troublesome for Mr. de Blasio, who argued his case from afar because of a limited number of allies in Albany, including an ambivalent Senate Republican majority that is aggrieved at past efforts by the mayor to have them voted from office.