"I'd like to have an open process. The only way I would filibuster a bill is if (Majority Leader Harry Reid, D-Nev.) did not allow alternative amendments," Graham said on CNN's "State of the Union."
His comments come after five Republicans joined in agreement to filibuster any gun legislation that may infringe on the Second Amendment to the U.S. Constitution.
Graham said he would vote against a measure that would require background checks for firearm purchases at gun shows, as well as a requirement for the sellers to keep the paperwork from such transactions.
"Before you'd expand the background check, there are 76,000 people, last year, failed a background check. And less than 1 percent got prosecuted ... . So, before you expand background checks to include private individuals, let's put some resources into the current system we have that's clearly broken," he said.
Graham said the overwhelming support for Americans on the expansion of background checks does not alter his opinion.
"The bill requires private individual transfers to go through the federal system. The current system is broken," he said. "I think that legislation is going nowhere."
Graham said he supports legislation to prevent the mentally ill from obtaining guns, and believes such legislation could be passed with a bipartisan approach.
Notable deaths of 2014 [PHOTOS]