Some of the measures under discussion are an increased police presence at home-district events, increased airport security for congressional members and a bullet-proof barrier in the House gallery, The Hill reported Tuesday.
Giffords was shot in the head Saturday during a political event at a grocery store in Tucson. She was among the 14 people wounded in the incident that left six other people dead, including a federal judge and a member of Giffords' staff.
U.S. Capitol Police officials and the sergeants at arms were expected to brief members of Congress at the Capitol Wednesday on how lawmakers can add more protection, focusing on closer communication with local law enforcement, law officials said. Providing one Capitol Police officer to each of the 435 lawmakers has been ruled unfeasible.
"I want to have a review from A to Z," said Rep. Dan Lungren, R-Calif., chairman of the House Administration Committee. "What are we doing? How are we doing it? How effective are we on it? Are we agile enough to respond to threats as we know them? And what do we do to give members of Congress that kind of information and that sense of appropriate awareness to protect themselves as best they can?"
Rep. Dan Burton, R-Ind., told The Hill he plans to re-introduce legislation that would create a barrier around the House gallery to protect members of Congress from attacks. Burton said his decision wasn't influenced by the Giffords shooting but it did point up the vulnerability of lawmakers.
"There are ways that someone can get an explosive device into the Capitol. If I know about it, then it seems that terrorists or others can figure it out without a great deal of trouble," Burton said.
While agreeing that heightened security must be considered, several lawmakers said decisions should throw up barriers between lawmakers and their constituents.
Senate Sergeant at Arms Terry Gainer said he heard similar sentiments from the Senate as well.
"The senators that I've talked to and the leadership does not want to be encumbered and be put into a cocoon," Gainer said. "They want to do their job and they want us to do everything we can to minimize the risk, and we're going to do that."
Notable deaths of 2014 [PHOTOS]