Sen. Al Franken wants Attorney General Jeff Sessions to come back and testify before the Senate Judiciary Committee after a report emerged Friday alleging that Mr. Sessions spoke to Russian Ambassador Sergey Kislyak about the Trump campaign during the 2016 election.

Mr. Sessions testified earlier this year that he didn’t speak with Russian officials about the campaign. In March he recused himself from involvement in the Justice Department’s investigation into allegations of collusion between President Trump’s campaign team and the Russian government during the 2016 election because he had served as a campaign surrogate at the time for Mr. Trump.

But a Friday report from The Washington Post, citing anonymous sources, said Mr. Kislyak spoke with the former Alabama senator about the campaign and reported that U.S. agents had obtained Mr. Kislyak’s communications.

“Chairman [Chuck] Grassley does want him to come back,” Mr. Franken, Minnesota Democrat, said on CNN’s “State of the Union,” referring to the Iowa Republican who heads the Senate Judiciary Committee.

Mr. Franken said he was unsure when or if Mr. Sessions would testify again before the Senate since the Judiciary Committee is first scheduled to hear from Donald Trump Jr. and former campaign manager Paul Manafort behind closed doors.

Mr. Franken said the two should testify under oath and release all requested documents to the committee after they reached a deal to avoid appearing in public to answer questions about a meeting they held last summer with a Russian attorney.

But Sen. Susan M. Collins, Maine Republican, said there is a big difference between alleged intercepts versus sworn testimony when asked about The Post’s report and Mr. Sessions.

“The Russian ambassador is our adversary, and the Russians have shown themselves to be masters at misinformation,” Ms. Collins said Sunday on “Face the Nation” on CBS.

Justice Department spokeswoman Sarah Isgur Flores has denied The Post’s report and said Mr. Sessions stands by his prior testimony.

But the top Democrat on the House intelligence committee, Rep. Adam B. Schiff of California, said he’s concerned about Mr. Sessions’ communications with Russian officials last year even beyond what The Post reported.

“There’s a lot about what the attorney general has said that just doesn’t hold much water,” Mr. Schiff said on “Face the Nation.”

Mr. Schiff’s committee is scheduled to hear from Mr. Trump’s son-in-law, Jared Kushner, during a private hearing on Tuesday after Mr. Kushner appears on Monday before the Senate intelligence committee to answer questions about the meeting with a Russian last June, which was also attended by Mr. Manafort and Donald Trump Jr.

Sarah Huckabee Sanders, the new White House press secretary, refused to discuss the anonymously sourced story from The Post, instead saying the number of leaks from the intelligence community is “one of the most undersold stories.”

“We’ve had over 60 leaks in the first six months in the intelligence community, when the other administrations previous to us were in the single digits,” Mrs. Sanders said Sunday on ABC’s “This Week.” “We need to focus on these leaks.”

Mrs. Sanders’ comment comes after Mr. Trump tweeted on Saturday about the “illegal leaks” and his attorney general.

“A new INTELLIGENCE LEAK from the Amazon Washington Post, this time against A.G. Jeff Sessions. These illegal leaks, like Comey’s, must stop!” said Mr. Trump.

The president’s tweet comes just days after Mr. Trump told The New York Times he wouldn’t have named Mr. Sessions as attorney general if he knew he would recuse himself from the Russia investigation.