President Obama’s homeland security chief on Wednesday told a House panel investigating Russian meddling in the November election that, much to his disappointment, Democratic National Committee officials declined an offer by his agency to help even after they learned the committee had been hacked.

In widely anticipated testimony before the House Permanent Select Committee on Intelligence, former Homeland Security Secretary Jeh Johnson provided new details about an unprecedented series of Kremlin-sponsored cyberattacks during the presidential campaign, while warning that such attacks “are going to get worse before they get better.”

At a separate hearing of the Senate Select Committee on Intelligence, Homeland Security and FBI officials said Russian government-linked hackers targeted as many as 21 states’ election systems last year, although there was no sign that any vote totals were changed or that the hacking affected the final result.

Mr. Johnson said his department identified Russian interference as “a front-burner item” last summer and reached out numerous times to assist the DNC after its network servers were hacked.

“I was anxious to know whether our folks were in there,” he said. “The response I got was FBI had spoken to them, they don’t want our help, they have [private cybersecurity firm] CrowdStrike. And that was the answer I got after I asked the question a number of times over the progression of time.

“I recall very clearly that I was not pleased that we were not in there helping them patch this vulnerability,” Mr. Johnson said.

The U.S. intelligence community believes sensitive information stolen from DNC servers was later passed to the international group WikiLeaks by foreign cyberoperatives, reportedly guided by Kremlin intelligence units. WikiLeaks has consistently claimed that the person who leaked the embarrassing data was a Democratic Party staffer who was unhappy about the party’s tilt against Sen. Bernard Sanders in the primary battle with Hillary Clinton.

DNC officials and Clinton campaign chairman John Podesta have in turn criticized the FBI, saying it made only perfunctory efforts to warn party officials about the depth and severity of the Russian hacking efforts.

“At no point during my tenure at the DNC did anyone from the FBI or any other government agency contact or communicate with me about Russian intrusion on the DNC network,” said a statement released in response by Rep. Debbie Wasserman Schultz of Florida, who was DNC chairwoman last year. “It is astounding to me that the chair of an organization like the DNC was never contacted by the FBI or any other agency concerned about these intrusions.”

Mr. Johnson also added fresh fuel to the scrutiny of fired FBI Director James B. Comey and his handling of the Clinton campaign email scandal during the presidential race. He raised questions about the delay from the time the DNC and FBI first discussed Russian hacking to the point where his department learned of the breach.

“The FBI and the DNC had been in contact with each other months before about the intrusion,” Mr. Johnson said.

Both Republicans and Democrats lined up to take shots at Mr. Johnson.

Rep. Trey Gowdy, South Carolina Republican, pointedly asked whether the former homeland security chief had any evidence of collusion between Moscow and the Trump campaign.

“Not beyond what has been out there open-source,” Mr. Johnson replied. “And not beyond anything that I’m sure this committee has already seen and heard before, directly from the intelligence community.”

Chaotic circumstances

The committee’s ranking Democrat, Rep. Adam B. Schiff of California, asked Mr. Johnson about the Obama administration’s slow public response to the Russia’s cyberattacks, which were first mentioned publicly in October. Mr. Johnson said he and other Obama administration officials faced a uniquely chaotic and complex set of circumstances as Election Day neared.

“This was a big decision, and there were a lot of considerations that went into it,” Mr. Johnson said. “This was an unprecedented step.”

The Obama White House, he said, feared that a public accusation of Russian interference could have been construed as an effort to influence the election. There was also the matter of Republican candidate Donald Trump’s heated rhetoric.

“One of the candidates, as you recall, was predicting that the election was going to be rigged,” said Mr. Johnson, adding that such a “statement might be seen as challenging the integrity of the process itself.”

Mr. Johnson added that timing was also an issue. The Obama administration released its public statement about Russian hacking on the same day as the release of an old video apparently capturing Mr. Trump bragging about groping women.

Separately, the Democratic and Republican leaders of the Senate Judiciary Committee met with special counsel Robert Mueller, who is investigating the Russian meddling scandal, suspected collusion with Trump campaign officials and, potentially, where the Trump White House tried to block the probe. The purpose of the meeting was to avoid clashes between Mr. Mueller’s independent probe and the various committees investigating aspects of the scandal on Capitol Hill.

In a separate Senate intelligence committee hearing on Wednesday morning, officials from the Department of Homeland Security and FBI gave new details on cyberattacks that they said targeted election systems in nearly two dozen unidentified states.

The email hacks that led to the release before the election of a trove of internal communications from the Democratic National Committee were all part of Russia’s “information warfare” campaign, said Bill Priestap, assistant director of the FBI’s Counterintelligence Division.

He told the Senate Select Committee on Intelligence that Russia also “pushed fake news and propaganda” to sow discord and undermine the key democratic principle of free and fair elections.

Voter information that was stolen through the election system hacks was likely taken in order to allow the Russians to better understand what kind of data were available, allowing more targeted intrusions on future elections, Mr. Priestap said.

The president-elect of the National Association of Secretaries of State, Connie Lawson, told lawmakers that state election officials were learning only now about the breadth and depth of the Russian hacking.

She pointed to recently leaked analysis from the National Security Agency, which found that Russian covert efforts continued much closer to Election Day. They included a “spear-phishing” attack led by Russian military intelligence on more than 100 local election officials just days before Nov. 8.

On another front, a group of House Democrats asked the White House to explain why security clearance for President Trump’s son-in-law and senior adviser, Jared Kushner, was not suspended after reports indicated that he failed to disclose his contacts with Russian officials and businessmen.

Democrats on the House Committee on Oversight and Government Reform wrote to White House Chief of Staff Reince Priebus on Wednesday requesting documents related to Mr. Kushner, including his security clearance application and any information related to classified information to which he has had access since December. The request also asks for similar information about former National Security Adviser Michael Flynn, who resigned after admitting he misled Vice President Mike Pence on the subject of phone calls Mr. Flynn placed to the Russian ambassador.

The lawmakers wrote that they were concerned about whether the White House was “properly safeguarding classified information” and sought documentation related to security clearance procedures.