Last week, congressional aides said the online giants were pushing back against lawmakers. The companies argued against the ad disclosure bill and dragged their feet to participate in a hearing on Tuesday. Lobbyists for Facebook explained how their voluntary efforts on political ad disclosures could work better than regulations.

Twitter has not addressed its problems with automated accounts, known as bots, that can quickly spread propaganda or fake news. In addition, not all social media companies have announced that they will tighten rules for buying political ads, one of the most pressing issues. And even if they do, lawmakers should still make more disclosure a legal requirement, said Senator Amy Klobuchar, Democrat from Minnesota, who is a co-sponsor of the disclosure bill with Mr. Warner and Senator John McCain, Republican of Arizona.

“If their policies comply with our bill, they should support our bill and maybe can use their policy as a standard when we pass our bill,” Ms. Klobuchar said.

Much of the harshest criticism so far is coming from Democrats like Mr. Warner and Ms. Klobuchar, which could limit the fallout for the tech companies. The Republican chairman of the Senate Intelligence Committee, Mr. Burr, has been more reserved than Mr. Warner, for example.

Many Republicans are reluctant to add any regulations to businesses. Some are also afraid of crossing other party members or President Trump, who regularly play down Russian interference in the election.

Republicans have slowly begun calling for more scrutiny of the companies. But lawmakers who are more oriented toward free markets have been reluctant to regulate the rapidly evolving industry. Other top regulators, particularly those appointed by Mr. Trump, are also considered unlikely to take strong action against the industry.