Republicans need to pick up four seats to claim control of the chamber, an attainable effort, some analysts said, because 23 of the 33 seats up for election are now under Democratic control, The Washington Post reported.
Republicans are favored by many experts to take Democratic seats in North Dakota and Nebraska and 10 or 12 other seats are considered in play.
A surprise announcement Monday by Rep. Jason Chaffetz, R-Utah, likely won't play into any potential shift, the Post said. Chaffetz said he wouldn't challenge Sen. Orrin G. Hatch for the party nomination. Polling indicated Hatch was vulnerable to a more conservative challenger like Chaffetz.
"Ultimately, I can spend the next 15 months doing my job or I can spend the next 15 months campaigning to do Senator Hatch's," Chaffetz said in a statement.
Rep. Allen West, R-Fla., said Monday he wouldn't try to challenge Democratic Sen. Bob Nelson, saying he wanted to remain in the House.
Former Rep. Chris Shays announced he would run for an open seat in Connecticut, trying to mount a political comeback after losing his 2008 re-election bid. He likely would face a primary with former World Wrestling Entertainment executive Linda McMahon, the GOP nominee for another open seat in Connecticut in 2010.
The Post said Democrats would be favored to retain the seat now occupied by Sen. Joseph I. Lieberman, an Independent, who is retiring.
The various Republican announcements came a few days after former Obama administration adviser Elizabeth Warren said she was forming an exploratory committee for a possible challenge to Sen. Scott Brown, R-Mass.