Also, Sen. Roy Blunt, R-Missouri, introduced the Retail Investor Protection Act, companion legislation to Rep. Anne Wagner's bill in the House that would require the Securities and Exchange Commission to be the lead regulator in writing a uniform fiduciary standard.

Read: Benefits Pro coverage on the DOL fiduciary rule

In a joint statement, Senate Republican sponsors and co-sponsors of the legislation said, "There is bipartisan agreement that we can raise standards without Washington dictating who Americans can hire and what investments they can make."

The companion House and Senate bills also include language that supporters say would adequately raise the bar for investment advice on IRAs and 401(k) plans.

But backers of the DOL say the Republican bills do not go far enough to protect investors from conflicted advice, which the White House has said costs investors billions of dollars a year.

During this week's mark up of The Affordable Retirement Advice Protection Act in a House Education and Workforce Committee hearing, ranking member Rep. Bobby Scott, D-Virginia, called the bill "constitutionally suspect," arguing that a requirement to put the DOL proposal to a Congressional vote violates the separation of power established in the U.S. Constitution.

Scott also said the fiduciary standards proposed in the legislation are too lax, and would allow unscrupulous advisors "to use fine print and boilerplate language" to "disclose and disclaim away their fiduciary obligation."

In the Ways and Means Committee's mark up of the SAVERs Act, three Democrats– Rep. Richard Neal, D- Massachusetts, Rep. Mike Thompson, D-California and Rep. John Larson, D-Connecticut–joined Republicans in moving the bill out of committee.

Throughout the debate, scores of Congressional Democrats have written Labor Secretary Thomas Perez, raising concerns that the DOL's proposal may adversely impact low and middle-income savers.

Those concerns included how the proposal may impact savers' access to annuity products, the distribution of proprietary investment products, whether or not participants in 401(k) plans could be provided thorough access to education, and whether or not the proposal's "seller's carve out," which requires advisors to 401(k) plans with fewer than 100 participants to act as fiduciaries, would not negatively impact small businesses.

Despite concerns raised by Democrats—one letter to Sec. Perez signed by 96 Democratic lawmakers in the House noted "practical problems" with DOL's proposal–Committee votes have attracted scant support from Democrats.

Some stakeholders have suggested that proves that a legislative effort to block the DOL's fiduciary rule would not survive a Presidential veto.

Lee Covington, general counsel at the Insured Retirement Institute, which lobby's on behalf of broker dealers and annuity providers, disagrees, and says the legislative efforts to counter the DOL are consequential, no matter their limited Democratic support thus far.

"These proposed laws send a strong signal to DOL that Congress wants to this rule to work, and they need to make the changes to the proposal that they've committed to," said Covington.

In not voting for legislation in this week's mark up hearings, Democrats are not necessarily voicing wholesale support for the DOL, but are instead saying they want to see what is in the rule first, said Covington.

"There has been an unprecedented amount of engagement on this issue from both sides of the aisle," he added. "If the DOL has not made significant changes to its proposal, I expect to see a groundswell of bipartisan support for a legislative alternative."

Jill Hoffman, vice president of government affairs at Financial Services Roundtable, another advocacy group for financial services interests, said, "Congress feels the need to be responsive on this issue."

"The substantive concerns are significant," said Hoffman, noting that several Democrats issued caveats in this week's Ways and Means Committee mark up hearing as to the extent of changes DOL has made to its proposal.

The Office of Management and Budget is now reviewing those changes, and is expected to release the final version of the rule by April.

Last year, two comment periods and a four-day open public forum yielded thousands of comments from stakeholders.

The DOL spent last fall and the early part of this year considering industry's suggestions to amend the proposal. Sec. Perez has made consistent assurances throughout this process that Labor intends to take seriously industry's suggestions, so as to write a "workable" rule.

But calls to open a third comment period, which would have made public the changes DOL has made to the rule, were denied.

Rep. Jared Polis, D-Colorado, ranking member of the House Subcommittee on Health, Employment, Labor and Pensions, has written OMB director Shaun Donovan this week, requesting that lawmakers in leadership positions with jurisdiction over retirement policy see the DOL's rule immediately.

In the letter, which was copied to Sec. Perez, Polis said he has a responsibility to "ensure the rule serves the interests of the American people."

"To fulfill that responsibility, my colleagues and I need the opportunity to see and review the rule as soon as possible," wrote Polis.

In her capacity at FSI, Hoffman wonders why Congress should not be allowed to see the amendments to the rule.

"Congress has oversight here," said Hoffman. "Its authority has been taken away by an administrative rule making process. Ultimately, Congress and the people they represent are going to have to live with this rule after 2016."

She added, "It is entirely appropriate for Congress to see this rule now."

Requests for comment from OMB and DOL were not returned.