But there are splits within the cabinet over the issue, and Mahmood is in the minority, BBC reported Monday evening.
Over 70 Labor MPs have also called on Starmer either to resign or to set out a timetable for his departure, while at least four Parliamentary Private Secretaries have resigned.
The prime minister is "weighing up his options" ahead of a cabinet meeting on Tuesday morning, according to Sky News, Xinhua reported.
Starmer is facing mounting pressure to resign following his party's disastrous performance in the local elections. Labor lost almost 1,500 council seats in England and surrendered control of around 40 local councils, including several traditional strongholds.
Speaking on Monday morning, Starmer said while the results were "very tough," he would not be "walking away."
Asked whether he would "fight" potential challengers within his party over his leadership, he replied "Yes," reaffirming: "I'm not gonna walk away."