WASHINGTON – Advocates for tightening the rules on stock trading among members of Congress gained steam Wednesday with the announcement of a fresh bipartisan Senate bill and Speaker Nancy Pelosi expressing an appetite for changes. But whether lawmakers will be prohibited from trading individual company stocks is still unclear.

“I’m a big believer in our committees and we’ve tasked the House Administration Committee to review the options that members are putting forth and they have different views on the subject,” Pelosi told reporters.

The California Democrat said she wanted to see certain criteria emerge from that process. That includes bolstering fines for noncompliance with the Stock Act and requiring financial disclosures in the federal judiciary. Financial disclosures need to be “government-wide,” she said. Aside from those two points, it is uncertain what will emerge from the House Administration Committee.

“But it’s complicated and members will figure it out and then we’ll go forward with what the consensus is,” Pelosi said.

A spokeperson for the House Administration Committee did not comment.

Senate Majority Leader Charles E. Schumer, D-N.Y., meanwhile said Wednesday he believes stock trading among members of Congress is an “important issue that Congress should address.” Schumer said he asked Democrats to produce a single bill the Senate can work on and that he hopes the chamber will “act on it soon.”

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The Stop Trading on Congressional Knowledge Act (STOCK) of 2012 was codified into law to prevent members from trading on insider information through disclosure requirements and forbade them from trading on nonpublic, material information gleaned as part of their jobs.

Members are required to disclose certain securities transitions over $1,000 made by them, a spouse or child no later than 45 days after the trade. Attention to the issue was magnified after Insider published accounts of lawmakers failing to comply with the Stock Act. The penalty for a late periodic transaction report is $200.

“We have to tighten the fines on those who violate the Stock Act,” Pelosi said, adding that it is currently “not sufficient to deter behavior.”

There are many bills seeking to address the issue. Reps. Abigail Spanberger, D-Va., and Chip Roy, R-Texas, have a bipartisan bill with more than 40 co-sponsors that would require lawmakers, spouses and dependent children to put specific assets – including individual company stocks, but not mutual funds – in a qualified blind trust. Republican co-sponsors include Reps. Andy Harris of Maryland, Scott Perry of Pennsylvania and Matt Gaetz of Florida.

Similar legislation in the Senate has been introduced by Sens. Jon Ossoff, D-Ga., and Mark Kelly, D-Ariz. Ossoff, in a news conference Wednesday afternoon, said he welcomes other proposals and feedback that he wants to gather by the end of the week.

“So there are some minor details to work out, but again, it’s going to be a five- or six-page bill. That’s why I want to get all this feedback in this week. Let’s see what other legislation is offered this week,” Ossoff said. “Then let’s come together, hammer out final bill text, move it to the floor expeditiously.”

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Some lawmakers have expressed hesitation on moving assets to a blind trust because there could still be some room for conflicts.

Sens. Steve Daines, R-Mont., and Elizabeth Warren, D-Mass., introduced a bipartisan bill Wednesday co-sponsored by Sens. Marsha Blackburn, R-Tenn., and Debbie Stabenow, D-Mich. Under the legislation, members and their spouses would be prohibited from owning an interest in or trading – except as a divestment – any stock, bond, commodity, future or other form of security. Lawmakers and spouses would be permitted to trade and hold mutual funds and exchange-traded funds (ETFs) as long as they do not present a conflict of interest and are diversified.

If the bill became law, current members and spouses would be required to divest those assets no later than 180 days after it took effect. New members and their spouses would have to divest by 180 days into that member’s first term. For complex investment vehicles, such as an interest in a hedge fund or venture capital fund, the divestment period would be five years.

The Department of Justice and the U.S. Office of Special Counsel would enforce the law and the penalty would be up to a $50,000 fine for each infraction.

“No one should ever have to wonder whether their Member of Congress is working for the public interest or their own financial interest. I’ve fought for years to root out corruption in Washington, and to ban federal officials from owning and trading individual stocks,” Warren said in a statement. “With Senator Daines and this strong coalition, I’m proud to introduce the first bipartisan bill in the Senate on this critical issue, and now we have a real chance to get it done.”

Reps. Pramila Jayapal, D-Wash., and Matt Rosendale, R-Mont., introduced a companion bill in the House.

Blind trusts are not mentioned in the legislation. Daines said in an interview with CNBC that they can present conflicts of interest.

“You put assets in a blind trust, you’re still aware of what assets are put in that blind trust, that’s just the reality of it,” Daines said. “And we’re trying to reduce the conflicts of interest here. There are no perfect solutions, but there are much better solutions and I think what Elizabeth Warren and I have proposed here … I think we’ve got something that can actually pass.”

Staffers and children of members would be able to trade under the bill. Regarding staffers, Daines told CNBC he doesn’t want “to prevent the best, the brightest, the most talented individuals from coming to serve here on Capitol Hill.”

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