The Senate could act as early as this week on a bill that would improve the nation’s background check system used for scrutinizing most firearms purchases, Republicans said Monday.

Senate Majority Whip John Cornyn, R-Texas, moved Monday afternoon to bring up for an immediate vote the Fix NICS bill, a bipartisan measure that would use both penalties and rewards to get more states and federal agencies to send information to the National Instant Criminal Background Check System.