Speaker John Boehner (R-Ohio) said he has "made clear if the Senate acts on gun control legislation, the House will consider it."
Boehner said lawmakers need to examine how to stop people with mental illnesses from obtaining firearms.
"And if you've looked at all these mass shootings, what you see is the people who perpetrated these crimes all had a history of mental illness," he said.
"So where's the nexus? And how do we ensure that we keep guns out of the hands of people who should not have them?"
Boehner said he expected House committees will continue exploring the issue.
Senate Democrats have introduced legislation which would ban the sale of assault weapons and high-capacity ammunition, but that bill is unlikely to pass in the House.
In addition, a bipartisan Senate group is working on legislation that would expand background checks on firearm purchasers.
The background-check proposal is seen as more likely to pull GOP support, but the talks have bogged down amid fears that universal checks could lead to the creation of a national database of gun owners.