MADISON, Wisconsin — An aide for Senator Johnson, Republican of Wisconsin, told Vice President Pence’s staff that the senator wanted to hand-deliver to Mr. Pence fake elector votes from his state and neighboring Michigan, according to text messages shared at Tuesday’s meeting of the House select committee investigating the January 6, 2021, Capitol riot.

A spokeswoman for Mr. Johnson, Alexa Henning, downplayed the texts after they were publicly disclosed for the first time during the committee’s hearing at Washington, but did not deny that Mr. Johnson had wanted to hand-deliver the slate of fake electors to Mr. Pence.

“The senator had no involvement in the creation of an alternate slate of electors and had no foreknowledge that it was going to be delivered to our office,” Ms. Henning tweeted. “This was a staff to staff exchange. His new Chief of Staff contacted the Vice President’s office. The Vice President’s office said not to give it to him and we did not. There was no further action taken. End of story.”

Mr. Johnson had planned to object to accepting electors in states where he said the results were in dispute. He signed on to the objection of Arizona’s electors, which happened before the attack, but then voted to accept those electors later. He did not object to accepting Wisconsin’s votes, but two of the state’s Republican congressmen did.

The evidence presented showed a back and forth between an aide to Mr. Johnson, Sean Riley, and a member of Mr. Pence’s staff, Chris Hodgson, at 12:37 p.m. on January 6, 2021.

“Johnson needs to hand something to VPOTUS please advise,” Mr. Riley said in a text to Mr. Hodgson.

“What is it?” Mr. Hodgson replied.

“Alternate slate of electors for MI and WI because archivist didn’t receive them,” Mr. Riley wrote back.

Mr. Hodgson responded: “Do not give that to him.”

Wisconsin Republicans met at the state capitol on December 14, 2020, the same day as 10 Democratic electors awarded their votes to President Biden, who carried the battleground state by just less than 21,000 votes. They forwarded their votes for President Trump to the National Archives, arguing that they were trying to preserve Mr. Trump’s legal options in case a court overturned Mr. Biden’s win.

Wisconsin’s bipartisan elections commission declined to punish the fake electors, determining that no election laws had been violated. The fake electors are now being sued in Wisconsin by a liberal law firm that is seeking $2.4 million in damages.

Other evidence shown at Tuesday’s hearing indicated that Mr. Trump’s campaign team wanted the Wisconsin fake electors to fly the paperwork to Washington.

“Freaking Trump idiots want someone to fly original elector papers to the senate President,” the executive director of the Wisconsin Republican Party, Mark Jefferson, texted on January 4. “They’re going to call one of us to tell us just what the hell is going on.”

Mr. Johnson met with Wisconsin lawmakers later in 2021 and talked about dismantling the state’s bipartisan elections commission and having the GOP-controlled legislature take over presidential and federal elections.

Mr. Johnson’s Democratic opponents jumped at the disclosure at Tuesday’s hearing and called on him to resign.

“Ron Johnson actively tried to undermine this democracy,” Lieutenant Governor Mandela Barnes said in a statement calling for Mr. Johnson’s resignation. “He literally tried to hand Mike Pence fake ballots. Once again, Ron Johnson has proven he’s a danger to our country and our fundamental rights.”

Other Democratic candidates, including the state treasurer, Sarah Godlewski, and the Outagamie County executive, Tom Nelson, made similar comments calling for his resignation. 

A Milwaukee Bucks executive, Alex Lasry, called Mr. Johnson a “seditious traitor and a danger to our democracy” but stopped short of calling for him to resign, saying the only way Mr. Johnson will leave is if he’s defeated in November.

Mr. Nelson, who had earlier called for Mr. Johnson to be subpoenaed to testify before the January 6 commission, said the disclosures “go beyond anything I could have imagined for how far Ron Johnson would go to overturn our Wisconsin election result.”

He called for the Justice Department to investigate Mr. Johnson.