The background check system is designed to prevent criminals, people with severe mental problems and others from getting guns.
Tuesday's meeting came five days after the panel approved Congress' first gun control measure since December's horrific shooting at a Newtown, Conn., elementary school that left 26 students and educators dead.
The initial bill, brought forward by the Judiciary Committee's chairman, Sen. Patrick Leahy, D-Vt., and others, establishes long prison terms for illegal gun traffickers and straw purchasers, people who buy a firearm for criminals or others forbidden to buy one.
Schumer's background check bill would exempt only a narrow range of transactions from the checks, such as transactions between immediate family members or weapons loaned temporarily during sporting events.
It would also renew the requirement that states and federal agencies report records on felons, people with major mental health problems, drug abusers and others to the federal background check system — something that many states and agencies do poorly.
Schumer had hoped to win GOP support for his measure, and he spent weeks bargaining with conservative Sen. Tom Coburn, R-Okla., who carries an A rating from the National Rifle Association. Those talks foundered.
Coburn's backing could have helped Schumer win support from other Republicans and moderate Democrats from states with large numbers of GOP voters — potentially crucial because the background check measure is likely to need 60 votes in the 100-member Senate. There are 55 Democrats, including two independents who usually side with them.
To pressure lawmakers, a dozen clergy members from Newtown collected 4,000 signatures of religious leaders from around the country on a letter asking them to support expanded background checks, an assault weapons ban and other restrictions. The letter was published Monday as an ad in the Des Moines (Iowa) Register and was addressed to Grassley. The group planned to run the ad elsewhere as well.