The Republican primary season kicks into high gear next month with a series of high-profile Senate contests that will help determine whether the GOP can take control of the upper chamber from Democrats in the midterm elections.

The May matchups — some of which could evolve into runoff elections — will show the strength of the GOP as it tries to hang on to seats in Nebraska, Georgia and Kentucky and to seize Democrat-controlled seats in North Carolina, West Virginia and Oregon.

“The GOP could potentially win the Senate in November, but the party could hurt its chances if primary voters make poor decisions in some upcoming primaries,” said Kyle Kondik, of the University of Virginia’s Center for Politics. “Several states with Senate contests in May feature some classic insider-outsider matchups, and national Republicans are hoping the insider/establishment candidates get the nominations.”

Republicans need to win six, net, to take the gavel away from Senate Majority Leader Harry Reid, Nevada Democrat.

In its latest breakdown, the Center for Politics predicts Republicans will gain four to eight Senate seats.

Democrats, meanwhile, see the Senate races in Kentucky and Georgia as their best opportunities to pick up seats. They hope Republicans field candidates cut from the same cloth as some of their 2012 and 2010 nominees, including Sharron Angle in Nevada, Christine O’Donnell in Delaware and Todd Akin in Missouri, who lost seats that some political observers felt Republicans would have won had they put forward better candidates.

But, with the Senate majority within reach, the Republican establishment has taken a stronger stand against conservatives and tea-party-fueled insurgents in primary races.

Indeed, Senate Minority Leader Mitch McConnell of Kentucky told The New York Times last month that Republican incumbents would “crush” their tea party rivals in the primary season.

“In 2010, they did not take the tea party seriously,” said Jennifer Duffy, of the Cook Political Report. “In 2012, they tried to work with them. They tried to nominate or get behind candidates that they could both agree on. And in 2014, they are just fighting back. They want the strongest general election candidate in place because they do have a shot at the majority.”

The first competitive primary is Tuesday in North Carolina. The winner of the four-way race will face off in the midterm election against Sen. Kay R. Hagan, one of the most vulnerable Democrats.

North Carolina Gov. Pat McCrory on Tuesday became the latest member of the Republican establishment to endorse state House Speaker Thom Tillis.

Polls show Mr. Tillis holds a double-digit lead over his closest rival, Greg Bannon, a former doctor who has won endorsements from Sen. Rand Paul of Kentucky and conservative columnist Ann Coulter.

Mr. Tillis will have to clear a 40 percent threshold to avoid a July runoff election, which would give his rivals a chance to rally around an alternative.

A week later, voters in Nebraska and West Virginia head to the polls in races that Republicans are favored to win.

The four-way nomination race in Nebraska has turned nasty, with the top competitors, former Secretary of State Shane Osborn and Ben Sasse, increasing their attacks in the run-up to the May 13 primary.

The winner is expected to roll to victory in the November election and succeed retiring Sen. Mike Johanns, a Republican.

Rep. Shelley Moore Capito, meanwhile, is positioned to win the Republican primary in West Virginia, where the retirement of Sen. John D. Rockefeller IV, a Democrat, offers a prime pickup opportunity for Republicans.

Six states, including Georgia, Kentucky and Oregon, have primary elections scheduled for May 20. Those states could fall into Republican hands if the election becomes a tidal wave.

In Kentucky, Mr. McConnell faces off against businessman Matt Bevin, a political newcomer who has won the backing of groups aligned with the tea party that have accused Mr. McConnell of abandoning small-government principles.

Mr. Bevin has painted the incumbent as a conservative sellout and a Washington insider who has lost touch with the people in his home state.

Political observers say Mr. McConnell is poised to win, putting him a step closer to a sixth term and perhaps the Senate majority leadership.

Mr. McConnell has been focusing on tearing down Democratic candidate Alison Lundergan Grimes, Kentucky’s secretary of state.

Ms. Grimes is tied with Mr. McConnell in polls and has the support of former President Bill Clinton, a close ally of her father, Jerry Lundergan, a prominent Democrat power broker in the state.

Republicans in Georgia also will head to the polls May 20 to decide who in a crowded field will run against Democrat Michelle Nunn, the daughter of former Sen. Sam Nunn, in the race to succeed retiring Sen. Saxby Chambliss.

To win the nomination outright, a candidate must receive at least 50 percent of the vote. Otherwise, a runoff election will be held July 22.

Polls show that David Perdue is leading the pack, followed by Reps. Jack Kingston and Paul Broun. Former Secretary of State Karen Handel and Rep. Phil Gingrey are bringing up the rear.

Nicholas Easton, a professor at the Columbus State University in Georgia, said the outcome will decide the contours of the general election fight.

“Nunn would greatly prefer to run against one of the two right-wing congressmen, Gingrey or Broun, as her basic strategy for purple-red state Georgia is to appeal to the middle because of her name, her association with [George H.W.] Bush’s Points of Light Foundation, which she is already touting in her TV ads, and her moderate record,” Mr. Easton said. “An extremist opponent would allow her to peel off enough centrist Republicans to pull an upset.”

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