As debate begins this week on sweeping firearms legislation, supporters are running into renewed concern from conservatives that the bill could lead to a gun “registry” despite adamant pledges to prohibit this.

Fear of a national registry has remained even after two influential senators last week proposed a compromise on background checks. The proposal, which will be voted on as an amendment, would expand background checks to gun shows and Internet sales while exempting personal transactions like those among family members.

The provision, meant to ease concerns about the checks becoming too pervasive, also included language to bar the creation of a federal registry.