Democratic Rep. Gabrielle Giffords also said in a statement released Tuesday that President Barack Obama will request $500 million in funding for border security.
Part of Giffords' district borders Mexico.
Republican Sen. John McCain of Arizona plans to introduce a proposal Tuesday afternoon to pay for putting 6,000 National troops at the border.
Arizona Gov. Jan Brewer's spokesman said the governor hadn't been told of the move prior to her office being contacted by The Associated Press and had no immediate comment.
In 2006, President George W. Bush sent thousands of troops to the border to perform support duties that tie up immigration agents, who would then have more time to arrest illegal immigrants. The troops didn't perform significant law enforcement duties.
The goal of the Bush plan was to have the troops at the border until thousands of new Border Patrol agents were trained. Those border troops cost $1.2 billion to work at the border.
At the time, some officials along the border said they experienced a drop-off in traffic after National Guard soldiers began to arrive there.
That program has since ended, and politicians in border states have called for troops to be sent there to curb human and drug smuggling and prevent Mexico's drug violence from spilling over into the United States.
After the Bush plan ended, a much smaller number of National Guard troops still remained at the border, helping federal and state officers with communications and anti-drug efforts.