Hoping to galvanize their base further, Republicans are casting the midterm elections as a golden opportunity to fire Senate Majority Leader Harry Reid, the Nevada Democrat who has served as lead blocker for President Obama’s liberal agenda on Capitol Hill and chief opponent of GOP legislation.

The attacks are being espoused inside and outside the Capital Beltway and could become staples of the Republican message as the party looks to pick up the six seats needed to take the Senate gavel out of Mr. Reid’s hands.

On Wednesday, Republican National Committee Chairman Reince Priebus described Mr. Reid as “dirty” and “unethical.”

The RNC circulated a memo that played up a Federal Election Commission inquiry into campaign cash that Mr. Reid reimbursed after using the money to purchase gifts for his granddaughter.

“We’re going to continue engaging voters who want nothing more than to stop Obama’s big government overreach and fire Harry Reid from his post as Senate majority leader after Senate Democrats paved the way for ObamaCare and stalled dozens of jobs bills sent over from the Republican House,” Mr. Priebus said.

The chairman of the Louisiana Republican Party, meanwhile, filed a complaint with the Senate Select Committee on Ethics, calling on lawmakers to investigate whether Mr. Reid improperly used taxpayer money to fund online attacks against the Koch brothers. The Kochs have invested millions of dollars into conservative causes, including ads against Obamacare in states with vulnerable Senate Democrats.

Mr. Reid’s office brushed off the attack, saying it showed “Republicans’ blind obedience to the shadowy billionaire Koch brothers.”

“Nothing says ‘Republicans are the party of the top one percent’ like lashing out with meritless complaints as a screen to defend the Koch brothers as they rig the system to benefit billionaires like themselves,” Reid spokesman Adam Jentleson said in a statement. “Senator Reid will continue to do everything in his power to hold the Koch brothers and their Republican enablers accountable for trying to tilt the playing field in favor of the wealthy and against the middle class.”

Mr. Reid has been a thorn in Republicans’ side, serving as Mr. Obama’s last line of defense on Capitol Hill.

The five-term incumbent has thwarted Republican attempts to repeal Obamacare and has pushed an election-year agenda that includes raising the minimum wage. It also aims to give members of his caucus some political cover for Election Day.

He also, as his spokesman’s statement suggests, has been on a crusade against David and Charles Koch, calling on Republicans to disavow the men and declaring, “I’m not afraid of the Koch brothers.”

Ford O’Connell, a GOP strategist, said Mr. Reid, a former boxer and ex-chairman of the powerful Nevada Gaming Commission, likely relishes being “best bodyguard that president Obama has.” He said Republicans are targeting the Nevada Democrat in hopes of sustaining the momentum they have built in recent months attacking Mr. Obama and his signature legislative achievement, the Affordable Care Act.

“You need another quiver in the arrow to keep the grass roots energized. Not to mention, it is one heck of a fundraiser,” Mr. O’Connell said. “Reid has been a check on the Republican House while simultaneously pushing President Obama’s agenda.”

On the campaign trail, Republican candidates are criticizing Mr. Reid.

Rep. Cory Gardner, who is vying to challenge Democrat Sen. Mark Udall in Colorado, has vowed to “make Harry Reid a footnote in history.”

Senate Minority Leader Mitch McConnell, Kentucky Republican, has released a television that says a vote for Democrat Alison Lundergan Grimes, the Kentucky secretary of state, is a “vote in favor of Harry Reid and Barack Obama’s anti-gun, anti-coal and anti-Kentucky agenda.”

Thom Tillis, the speaker of the North Carolina House and front-runner for the Republican nomination for Senate, has cast Sen. Kay R. Hagan, a Democrat, as nothing more than a rubber stamp for Mr. Reid.

“A senator that votes 96 percent of the time with Harry Reid is not representative of North Carolina,” Mr. Tillis said. “She needs to be sent home.”

Jim Manley, a Democratic strategist and former aide to Mr. Reid, said he doesn’t think Republicans will get much mileage out of the attacks. He said his former boss is not as well-known as House Speaker Nancy Pelosi, the Californian to whom Republicans linked vulnerable Democrats, with some success, in previous elections.

“Very few Americans know who Sen. Reid is and so if they are going to go after him, I don’t think it is going to work,” Mr. Manley said. “It sure seems to me that the RNC is being awful sensitive these days and Sen. Reid is getting under their skin if they are going out and criticizing him like that.”

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