The Republicans hope to overturn the 55-45 Democratic majority in the Senate in the midterm elections. Republican control would mean the end of Reid's days as majority leader.
Speaking on the lawn on the East Front of the U.S. Capitol, several GOP lawmakers said Sen. Reid of Nevada is ignoring House-approved bills favored by Republicans.
"It is time to fire Harry Reid," Rep. Marsha Blackburn, R-Tenn., said. "We are ready to take control of the Senate."
GOP-sponsored bills were stacked several feet high on a round, mahogany table to the left of the lectern, showing the small crowd of Republican supporters the House-approved legislation that GOP members claim is "sitting on Harry Reid's desk."
"The American people are not getting an up-and-down vote in the United States Senate because Harry Reid is blocking that," said Rep. Steve Daines, R-Mont. "We fired Nancy Pelosi in 2010. It's time to fire Harry Reid in 2014."
Among the Congress members who spoke were Rep. Doug Collins, R-Ga., Rep. Bill Johnson, R-Ohio and Rep. David McKinley, R-W.Va.
Later Tuesday at the Capitol, Sen. Reid fired back.
"One thing the American people are totally focused on is how absurd the Republicans are spending their time, talking about impeachment and suing the president," he said. "This is a degree higher than absurdity."
When asked about the GOP presser about him, Reid quipped, "I'm surprised more press didn't show up. Only a couple people did."
The Republican National Committee started its "Fire Reid" campaign last week. As part of the movement, the RNC announced it launched robocalls in Alaska, Colorado, Iowa, Louisiana, Michigan, Montana, New Mexico, North Carolina, Oregon, South Dakota, Virginia and West Virginia — all states that have at least one senator up for reelection in November. The message urges the listener to vote against the Democratic Senate candidate in one of those states.
"What you didn't hear [today] is anything on some of the big issues Americans want Congress to address," said Michael Czin, press secretary for the Democratic National Committee. "Republicans are focused on mechanics, but they're not focused on passing common sense immigration reform."
The Republican-controlled house has not acted on the comprehensive immigration reform bill that passed in the Senate last year.
"They [Republicans] need to take some of their own advice," Czin said.
Republican interns and staffers, who made up most of the crowd, held up signs that read, "Stop Obama, Fire Reid." A banner with the same message was displayed behind the Republican congressional members.
While specifically targeting the 74-year-old Senate majority leader, the broad theme of the movement is to not only attack Reid, but also President Obama and the Democrats as a whole. Independent Sen. Bernie Sanders of Vermont sits with the Democratic caucus, providing the 55-vote majority.
"The only way to stop President Obama's agenda and to get jobs bills out of Congress is to fire Harry Reid," RNC chairman Reince Priebus said in a release. "And the way to fire Harry Reid is to support your Republican Senate candidate. It doesn't matter who the Democrat candidate is. A vote for a Democrat is a vote for Harry Reid."
Congress is set to go on a five-week recess at the end of the week.