“I don’t know that our movement is so divided — I don’t know that the president and I differ on issues, but we may differ on focus,” Mr. Pence said. Mr. Trump and Mr. Pence have been in somewhat regular contact since leaving office, speaking several times by phone in conversations that avoided discussing the insurgency on Capitol Hill, according to their advisers. In an interview last year, Mr. Trump said he never told Pence he was sorry for not acting more quickly to stop the attack — and that Pence never apologized. But a rivalry has been brewing behind the scenes. On Monday, Mr. Pence announced that his book about his time in administration, “So Help Me God,” will be published on Nov. 15. The book has caused tension with Mr. Trump, who, when he learned in early 2021 that Pence had a book deal, was still considering getting one of his own. But in most parts of the publishing industry, Mr. Trump was generally seen as a risk. The former president seemed disappointed that Mr. Pence had struck a deal and within days of learning, attacked the former vice president while speaking to a group of Republican donors at a Republican National Committee event in Mar-a-Lago, taking about Mr. Pence’s refusal to do what Mr. Trump wanted on January 6, 2021. The two men’s paths also diverged this year during the midterm campaign. They have backed opposing candidates in several primary races, including the Republican gubernatorial contest in Arizona, which will be decided next week, and the party’s gubernatorial primary in Georgia, where Mr. Pence’s pick, Gov. Brian Kemp, defeated easily beat Trump-endorsed challenger David Perdue.