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"I read Senator Shaheen's letter today and support the common sense idea to extend the date for open enrollment," Pryor says in a statement.

"I believe, given the technical issues, it makes sense to extend the time for people to sign up. In addition, the administration should state clearly how the enforcement mechanism will work if people can't sign up in time. We all want to see the law work, and I hope the administration will take a hard look at this reasonable suggestion."

Pryor is in a tight race with Rep. Tom CottonTom Bryant CottonOvernight Defense: Lawmakers release compromise defense bill in defiance of Trump veto threat | Senate voting next week on blocking UAE arms sale | Report faults lack of training, 'chronic fatigue' in military plane crashes Compromise defense bill excludes competing nuclear testing language Republican senators introduce bill to protect government workers from being targeted at home MORE (R-Ark.) in a state that heavily favors Republicans.

Begich, who is another GOP target in 2014, said he also supports a longer enrollment period.

“I have repeatedly said this law is not perfect and have proposed changes to make it work for Alaska families and small businesses. That's why I supported repealing the 1099 requirements and delaying the employer mandate," Begich said. 

"Given the recent website issues, I also support extending open enrollment season. I want to work with the administration to ensure that individuals are not unfairly penalized if technical issues with the website continue.”

Shaheen is also up for reelection in a swing state, though she has yet to draw an opponent and appears to be in good shape for reelection.

Sen. Joe ManchinJoseph (Joe) ManchinSmall businesses don't need another stimulus — they need customers Congress faces late-year logjam Funding bill hits snag as shutdown deadline looms MORE (D-W.Va.), a conservative Democrat not facing reelection, is pushing for a one-year delay of the individual mandate.

The government's federally run healthcare exchanges have continued to be beset by extensive technical failures, leading to increased worries from some politically vulnerable Democrats that the law will hinder their chances at reelection.

— This story was updated at 7:04 p.m.