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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.”