One approach designed to entice lawmakers representing large rural areas, particularly in Alaska, would exempt residents who live hundreds of miles from a gun dealer. Lawmakers are hoping that they can attract support from both Republicans and Democrats who are weighing the political costs and benefits of a bill against the perception that they are chipping away at gun rights.

Headed into votes this week, Senate Republicans and Democrats continued to negotiate how the legislation and its amendments, including the Toomey-Manchin measure, will be brought to the floor.

Some senators are advocating voting for a series of amendments in advance of the background check measure, allowing members to vote for or against proposals like a renewed assault weapons ban, a limit on high-capacity magazines and a new mental health provision before weighing in on the background check measure. Republicans are seeking their own amendments, including a measure that would make it easier to carry concealed weapons across state lines.