The House committee investigating the Jan. 6 attack on the Capitol has subpoenaed Mr. Jones, who echoed Mr. Trump’s lie that the 2020 election was stolen, and is scrutinizing his role in organizing “Stop the Steal” events.

Late Tuesday, Mr. Jones’s lawyers filed an “offer of compromise” in Connecticut, proposing to pay the Sandy Hook plaintiffs a settlement of $120,000 each. “Mr. Jones extends his heartfelt apology for any distress his remarks caused,” they said.

The families swiftly rejected the proposal. “The so-called offer is a transparent and desperate attempt by Alex Jones to escape a public reckoning under oath with his deceitful, profit-driven campaign against the plaintiffs and the memory of their loved ones lost at Sandy Hook,” the families said in their court filing rejecting the offer.

The families in the Texas cases, in which Mr. Jones has been deposed, also say they are eager for their day in court.

“I’m looking forward to staring him down in a courtroom, to remind him of how he threw salt in my wound,” Lenny Pozner, whose son Noah died at Sandy Hook, said in an interview. Mr. Pozner and his nonprofit, the HONR Network, have led efforts to hold Mr. Jones to account for his falsehoods about the shooting. “He’s been hung by his own actions,” Mr. Pozner added.

Chris Mattei, a lawyer on the families’ legal team in Connecticut, would depose Mr. Jones. Mr. Mattei said the lawyers also plan to depose Rob Dew, a top Jones lieutenant, and finish questioning several Infowars representatives. Mr. Jones’s father, David Jones, who bankrolled his son’s business at its founding and is deeply involved in its product sales, has so far evaded a subpoena seeking his deposition, Mr. Mattei said.

Of Mr. Jones’s many maneuvers in the run-up to trial, Mr. Mattei said the most egregious was “his fabrication and concealment of financial records” and failure to produce comprehensive data on traffic to the Infowars website.