The head of the local Democratic Party in Queens, where Assemblyman Zohran Mamdani got his political start, has never met him. The party’s longtime state chairman had not spoken to him until the day after a stunning primary night that stamped him as a rising Democratic star.

And among the party’s strategists, officials and elected leaders in Washington, he’s almost entirely unknown.

Now, as the Democratic nominee for mayor of New York City, Mr. Mamdani, 33, is on a crash course to change that. He is unleashing a full-scale charm offensive of private meetings, phone calls and public promises aimed at wooing top party leaders, donors and activists.

On Monday, he met with Jewish elected officials in New York City. The next day, he took pointed questions about his views on Israel and tax policy from a group of 150 business leaders in the city.

A day later, he headed to Capitol Hill to offer campaign advice to dozens of Democratic members of Congress at a breakfast hosted by Representative Alexandria Ocasio-Cortez of New York before returning to Manhattan for a private meeting with younger technology executives. And on Friday, he met with Representative Hakeem Jeffries, the House Democratic leader and a fellow New Yorker who has yet to endorse his bid.

Already, some establishment Democrats have been grappling with Mr. Mamdani’s sudden standing as their party’s standard-bearer in America’s most populous city. And they want to quickly get the measure of a man who has spent much of his political life far outside of their big tent.