WASHINGTON—The Orlando mass shooting sparked a new push in the Senate on Wednesday to reconcile measures from Democrats and Republicans aimed at restricting suspected terrorists’ access to weapons.

Senate Democrats, frustrated by congressional inaction in the face of a string of recent shootings, intensified pressure on Republicans to pass new restrictions. The impasse over how to respond to gun violence has bled into battleground Senate races and left House Democrats so incensed Monday evening that some refused to participate...