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The group made the recommendations this week to Vice President Joseph R. Biden Jr. and plans to release them publicly Friday afternoon. The Times obtained a copy of the document, which stresses that "closing the massive hole in the background check" system is the group's top policy priority.

"Calling it a 'gun show loophole' trivializes the problem," the document provided to Mr. Biden says. "Universal background checks on all gun sales would have a clear positive impact on public safety, and is also clearly compatible with the rights of law-abiding citizens to own guns."

Dan Gross, the group's president, said he and other advocates still feel strongly about the need to limit the availability of military-style assault weapons. The group said in its recommendations to Mr. Biden that any proposals should find ways to "limit the availability of military-style weapons and high-capacity ammunition that are designed for mass killing."

But Mr. Gross said that the group does not want a debate that focuses primarily on a possible ban of assault weapons.

"We're not rating any solution as bigger or more important than the other. But it's vitally important that the conversation be broader than just an assault ban," Mr. Gross said. "Background checks clearly stake out a middle ground that can save lives.

An even stronger message comes from Third Way, a left-leaning research group in Washington D.C., that has long advocated for stronger gun control laws. Matt Bennett, the vice president for public affairs at Third Way, said Friday that President Obama should not get into a knock-down fight with the National Rifle Association over an assault-weapons ban.

"The assault-weapons ban is a low priority relative to the other measures the Biden Task Force is considering," Mr. Bennett said. "Political capital in the gun debate only goes so far. We think it should be spent on things that would have the greatest impact on gun violence, like universal background checks and cracking down on gun trafficking."

The caution about the politics of fighting for an assault-weapons ban from some advocacy groups comes as Mr. Biden signaled his interest in focusing on other parts of the gun debate. In public comments Friday, Mr. Biden did not mention the idea of a gun ban.

The White House says President Obama still supports a ban on assault weapons and fully intends to propose and fight for one as part of a broader package of changes. But they acknowledge that the political fight will be difficult, especially in the Republican-controlled House.

Some advocacy organizations are still pushing for a sustained effort to revive the assault-weapons ban, which was initially imposed in 1994 and expired a decade later. But Mr. Bennett said his group was concerned that a prolonged and difficult fight with the N.R.A. over such a ban would make it more difficult to achieve the other parts of the plan.

"While they won't admit it, the N.R.A. probably wants the A.W.B. fight," Mr. Bennett said, "because it will dominate the debate, drive away some moderate members of Congress, and put the focus on a gun ban, rather than their outrageous and indefensible opposition to background checks, a modern gun trafficking law, and the use of gun violence data by law enforcement."

In addition to the background checks, the Brady campaign said there are several other actions that the government should take, including: Improving the ability to identify dangerous people who present the most risk, especially among the mentally ill; Supporting law enforcement by giving them the tools to crack down on gun trafficking and prevent straw purchases and eliminating legal loopholes that unfairly protect dangerous practices that contribute to gun deaths and injuries.

The group also said that "public education campaigns are critical to inspiring law-abiding individuals to make safer choices around gun ownership and access."