Mr. Bloomberg’s endorsement of a federal licensing policy is likely to ripple widely in the world of gun-control advocacy, where many groups, including those funded by Mr. Bloomberg, have focused chiefly on tightening background checks. The idea of a national licensing requirement, which would be even more restrictive, gained wide traction in the presidential race after Senator Cory Booker of New Jersey proposed one in May. At the time, John Feinblatt, a prominent gun-control strategist close to Mr. Bloomberg, expressed hesitation about the policy and suggested it was not “research tested .”

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But versions of the policy were quickly endorsed by leading candidates such as Senators Elizabeth Warren of Massachusetts and Bernie Sanders of Vermont and Mayor Pete Buttigieg of South Bend, Ind. Mr. Biden has not gone as far, proposing instead that states be encouraged to set up licensing systems but suggesting that it would be excessive and politically risky for the federal government to get directly involved in issuing gun licenses.

The stakes for Mr. Bloomberg in the gun debate may be particularly high, as he seeks to persuade Democratic voters to look past some of his more conservative positions — including his record of championing aggressive policing as mayor, and his general sympathy for the banking industry — and embrace him as a key champion of a few core progressive concerns. And as other Democrats criticize the way he is using his personal fortune in the election, Mr. Bloomberg has pointed to his self-funded advocacy on guns, climate and other issues as proof of his good intentions.

A number of the gun-control ideas Mr. Bloomberg is proposing have been debated extensively in Congress, endorsed by other Democratic presidential candidates and, in some cases, enacted at the state level by Democratic-controlled governments. His policy paper identifies states that have tried out some of the ideas he is endorsing, including red-flag laws, certain kinds of background checks, the elimination of gun-sales loopholes and different systems for tracking firearm sales.

Colorado has been a proving ground for some of those policies: As the state has trended steadily toward the Democrats, lawmakers there have passed laws tightening background checks, limiting the size of ammunition magazines and enacting red-flag procedures for seizing firearms from certain people. But the implementation of certain gun restrictions remains contested, with conservative Colorado sheriffs resisting them.

Mr. Sullivan, who said he had been briefed on Mr. Bloomberg’s plan, described it as mainly containing “the standard issues” that gun-control groups advocate, with the notable addition of the national licensing policy. That measure, he said, was “the next heavy lift” for lawmakers like him.

“It would be tough to do, but I think we’re headed that way,” he said.

Mr. Bloomberg’s agenda includes a long list of other restrictions on firearms and ammunition, including raising the minimum age for gun purchasers to 21, banning guns in schools and creating a safe-storage requirement for gun owners. He is proposing to appoint a White House gun coordinator and to declare gun violence a public health emergency, and to empower the Consumer Product Safety Commission to oversee guns as it does other products.