Mr. Tusk said he would seek to expand turnout in the Democratic primary, which Mr. de Blasio won with roughly 280,000 of 690,000 ballots cast in 2013, out of around three million registered Democrats. Mr. Tusk planned to court younger New Yorkers with the sorts of sharing apps and services that have become ubiquitous in many of their lives, including Airbnb, Uber and the restaurant delivery service Seamless. (None of the companies have yet been approached by Mr. Tusk about participating in a get-out-the-vote campaign, though he said he had discussed the idea with Uber at the national level when Mr. Bloomberg was mulling his presidential run.)

The poll, he said, would lay the groundwork for an as-yet-unnamed challenger. Among those included in the poll were a few familiar potential candidates — the public advocate, Letitia James; the comptroller, Scott M. Stringer; Representative Hakeem Jeffries — and some unusual ones like the Rev. Al Sharpton, the former police commissioner Raymond W. Kelly and the MSNBC host Joe Scarborough.

Another potential candidate, Paul J. Massey Jr., a real estate developer who has said he is mulling a Republican run for mayor, closed a nonprofit issue advocacy group he formed this year, 1NY Together, before it ever got off the ground — an indication of the shifting attitudes toward the nonprofit model that Mr. de Blasio had embraced. (Citing the completion of its work, Mr. de Blasio has said the Campaign for One New York has also stopped operations.)

“Too many 501(c)(4)’s have crossed the legal line, and I wouldn’t want to be mentioned in the same sentence with any of them,” he said, referring to the section of the tax code that allows for their creation.

Faced with the opposition, the mayor’s strategy has been threefold, aides said: Stay focused on the job; keep in constant communication with supporters and allies; and get the message out on the radio, in town-hall-style meetings and through community events.

In recent weeks, council members have been encouraged by the mayor’s office to post messages on Twitter vowing to “#protectprogress”; eight of 51 had done so by Friday. Unions, too, were pressed to express support.

“Screaming headlines have jumped ahead of the appropriate legal scrutiny now taking place,” Henry Garrido, the executive director of District Council 37, the city’s largest union of municipal workers, wrote in an email to his members on May 26. “I think it’s important during this challenging time to take stock of the accomplishments of the past two and a half years.”