The memorandum said that more than 800 mayors in the United States “support more aggressive prosecution of those who fail background checks.”

“This is not like looking for a needle in a haystack — these are people you know are too violent to buy a gun,” John Feinblatt, an official with the mayors’ group who met with Mr. Biden on Wednesday, said in a telephone interview. “Once they have been rejected, they go online or to a private seller or a gun show and get a gun.”

The low number of prosecutions in 2010, the most recent year for which data are available, is consistent with other years. Prosecuting these cases has proved challenging because to get a conviction “you have to prove that the person knew they were lying when they tried to purchase the firearm,” said a senior Justice Department official, who spoke on the condition of anonymity to discuss matters related to gun control before Mr. Biden’s proposals are announced.

A conviction usually carries a maximum sentence of just six months, the official said, adding that with a limited number of federal prosecutors the government has to prioritize its use of resources.

Although gun control advocates have been more vocal about the issue, the N.R.A. also supports similar action, arguing that the administration should enforce the gun laws that already exist before making new ones.

“It has been a longstanding frustration of the National Rifle Association that there’s no follow-through or follow-up on these cases and criminals, and those who shouldn’t be trying to buy guns have been getting away scot-free,” said Andrew Arulanandam, an N.R.A. spokesman.

Mr. Arulanandam said that the N.R.A. has “for decades been trying to get prior administrations — Republicans and Democrats — to take action on the matter but there seems to be no will by the Justice Department to enforce existing gun laws.”