With little hope of a quick resolution to the federal government shutdown, and as polls show voters largely blame the GOP for the impasse, Republicans could find themselves jeopardizing their hopes of winning control of the United States Senate next year.

That’s because, for the second electoral cycle in a row, Republicans are banking on a large number of House members running for Senate seats in key states. And, for the second cycle in a row, House Republicans are the most unpopular subset of a deeply unpopular Congress.

Republicans are likely to nominate a sitting House member for Democratic-held Senate seats in four states. Rep. Tom Cotton (R-Ark.) has no real opponents in his bid for the Republican nomination in the contest against Sen. Mark Pryor (D). Rep. Shelley Moore Capito (R-W.Va.) is the front-runner in the race to succeed retiring Sen. Jay Rockefeller (D). Rep. Bill Cassidy (R-La.) leads a wealthy self-funder in the race to face Sen. Mary Landrieu (D). And Rep. Steve Daines (R-Mont.) is likely to join the race to replace retiring Sen. Max Baucus (D) in the coming weeks.

And in Georgia, three House members — Reps. Phil Gingrey, Jack Kingston and Paul Broun — are among the half-dozen serious contenders for a seat being vacated by retiring Sen. Saxby Chambliss (R).

All seven have voted with Republicans on continuing resolutions that then failed in the Senate, as well as on piecemeal measures to fund particular functions of the government. None is among the 19 House Republicans who say they would support voting on a so-called “clean” continuing resolution.

Most of the seven have taken a similar line: that they oppose a shutdown, but that the Democratic-run Senate is to blame for inaction on the House-passed continuing resolutions.

“I never wanted it to come to this,” Capito told the Charleston Daily Mail on Wednesday. “But we couldn’t get the Senate to act.”

“Tom never advocated a government shutdown — that is something being promoted by Mark Pryor and Barack Obama,” Cotton spokeswoman Caroline Rabbitt told the Associated Press in late September. “Tom Cotton and Republicans are trying to reduce spending and spare Americans the train wreck that is Obamacare.”

“We’re hearing from the voice of Montanans and Americans, saying we want two things: We want to see the government remain open and have Congress act on some of the harmful provisions of Obamacare,” Daines told the Billings Gazette earlier this week. “I’m hoping the president and [House Speaker John Boehner] can engage so we can find some common ground to move forward.”

The only votes on which any of the seven Republicans have differed was on a version of the continuing resolution on Monday that included a one-year delay for the individual health-insurance mandate. Gingrey and Broun voted against the measure, along with a cohort of very conservative House Republicans and three moderates; the conservatives voted against the bill because they favored a repeal, rather than a delay. Kingston and the other four Senate candidates voted with the majority of their party.

Despite GOP efforts to place blame for the shutdown on the Senate, polls show pluralities of Americans blame it on Republicans. A CBS News poll conducted on Tuesday and Wednesday among 1,021 adults showed 44 percent blamed Republicans in Congress for the shutdown, while 35 percent said Democrats and President Obama were more to blame. A CNN/ORC International poll [pdf] conducted among 803 adults Sept. 27-29, before the shutdown, found that 46 percent would blame Republicans in Congress if a shutdown eventuated, while 36 percent would blame Obama. In both surveys, roughly 15 percent of Americans said they would blame both Republicans and Democrats equally.

Almost seven in 10 respondents told CNN pollsters they believed Republicans in Congress were acting like “spoiled children” in the debate over the federal budget. About half of Americans, 49 percent, said Obama was acting like a responsible adult; 47 percent said he was acting like a spoiled child, and 58 percent said the same about Democrats in Congress.

Meanwhile, the Republican Party’s approval rating has fallen to its lowest ebb since 1998, at the height of the Clinton impeachment drama. Just 32 percent of Americans told CNN they had a favorable impression of the GOP. Democrats don’t have a lot to brag about — only 43 percent said they viewed Democrats favorably, the lowest since 44 percent said the same just two months before the 2010 elections — but Republican numbers are consistently worse.

Democrats have had success pillorying House Republicans as the problem with Washington. In 2012, Obama made Republican obstructionism a pillar of his reelection campaign. That year, seven current or former House Republicans ran for Senate seats; only one, Arizona’s Jeff Flake, won his race.

Perhaps more ominously for the GOP, three House Republicans — Reps. Todd Akin (R-Mo.), Rick Berg (R-N.D.) and Denny Rehberg (R-Mont.) — lost to Democratic candidates in states that Republican presidential nominee Mitt Romney won.

“Voters, even in Republican-leaning states, do not want to elect partisan warriors posing as candidates who put their party politics ahead of what’s best for their constituents, and right now that is all the GOP is offering,” said Matt Canter, a spokesman for the Democratic Senatorial Campaign Committee.

Republicans hope that the shutdown will be bad for incumbents in general — and in Senate races, Democrats such as Pryor, Landrieu and Sens. Kay Hagan (D-N.C.) and Mark Begich (D-Alaska) are the incumbents. And voters in red states like Arkansas, Louisiana and West Virginia are more likely to place the blame for a shutdown at Democrats’ feet.

“Weak incumbents like Mary Landrieu, Mark Pryor, Kay Hagan and Mark Begich are the status quo in Washington and are forced to defend a system that is clearly isn’t working under their management,” said Brad Dayspring, a spokesman for the National Republican Senatorial Committee.

Then again, with Election Day 13 months down the road, there’s no guarantee that voters will have the shutdown in mind next November. The last time the government shut down, in 1995 and 1996, few competitive Senate contests later that year hinged on the shutdown. Republicans picked up Democratic-held seats in Alabama, Arkansas (even as Bill Clinton won his home state’s electoral college votes) and Nebraska, but the only incumbent to lose reelection was South Dakota Republican Larry Pressler.

And if the 2014 elections turn into a wave, Republican House members would be on the winning end. In 2010, when Republicans won big in House and Senate elections, six of seven sitting or former House members won Senate seats; only Rep. Mike Castle (R-Del.), who lost a Republican primary to tea party favorite Christine O’Donnell, fell short. And three Republicans, Rep. Mark Kirk (R-Ill.) and former representatives Pat Toomey (R-Pa.) and Rob Portman (R-Ohio), won seats in states Obama won in both 2008 and 2012.

More than half of the Senate’s members have served in the House. Twenty-six of the former House members are Democrats, while 24 are Republicans; Sen. Bernie Sanders (I-Vt.) also served in the House.

But there’s no question Republicans are relying on House members in critical seats on their path to a majority this year. If voters hold congressional Republicans responsible for the government shutdown next November, the GOP’s hopes of winning back the Senate would have faded this week.