To become law the measure must also be approved by the Senate and then signed by the president. If approved, it would become the US's first new gun control legislation since 1994.
The bill was drafted after Cho Seung-Hui, a student, shot and killed 32 students and faculty on the Virginia Tech campus on April 16, before turning the gun on himself.
Nancy Pelosi, the House speaker, said in a statement there was "an urgent national need to improve the background check system".
Background checks
 Pelosi said that "serious gaps exist" in the
current  gun control system [GALLO/GETTY] 
While a background check system is already in place, Pelosi said that "serious gaps exist" in the current system, allowing thousands of people "including those who have been adjudicated as mentally ill, to escape proper background checks".
George Bush, the US president, said he was "closely following legislative efforts to strengthen the instant background check system".
The government report, meanwhile, called not for tighter gun control laws but instead for increased sharing of information about students.
After the Virginia Tech shootings in April, Bush commissioned the report from Alberto Gonzales, the US attorney general, along with Michael Leavitt, health and human services secretary, and Margaret Spellings, the education secretary.
Their report, released on Wednesday, said that "confusion and differing interpretations about state and federal privacy laws" meant that information about "potentially dangerous" people had not been adequately shared.
Leavitt told reporters that a "key finding" of the study was "that in only about 23 states is information actually being reported to the instant gun check registry".
Groups supporting those with mental illness called the report "disappointing" and said that it ignored the issue of funding for their work.
Michael J Fitzpatrick, executive director of the National Alliance on Mental Illness, was quoted in the Washington Post as saying: "We don't need any more commissions or task forces. We know what to do ... the real problem is that help often is not available."
The report has also raised concerns over how to protect people's right to privacy, especially of those deemed mentally ill.