House Speaker Nancy Pelosi (D-Calif.) said in a statement after the Senate vote "the Rules Committee will meet to advance this life-saving legislation to the Floor" first thing on Friday morning.
"When the Rules Committee finishes its business, we will head immediately to the Floor," she added.
It also includes increased funding for mental health and school safety and incentives for states to implement "red flag" laws.
A bipartisan group of senators drafted the legislation in response to multiple mass shootings last month, including one at an elementary school in Uvalde, Texas
The bill was hung up last week as the "core four" senators involved in drafting the bill struggled to strike a compromise over the red flag provision and the "boyfriend loophole."
McConnell earlier this week called the bill "a common-sense package of popular steps that will help make these horrifying incidents less likely
"Families in Uvalde and Buffalo — and too many tragic shootings before — have demanded action. And tonight, we acted," Biden said.
"This bipartisan legislation will help protect Americans. Kids in schools and communities will be safer because of it.
The House of Representatives should promptly vote on this bipartisan bill and send it to my desk."