In a stop in Arizona, a key battleground state in next year’s election, the president plans to embrace the longtime Republican senator and vocal Trump critic.President Biden plans to announce on Thursday that he will devote federal money to create a new library and museum dedicated to his old friend and adversary, Senator John McCain, seeking to embrace a Republican who stood against former President Donald J. Trump.After stops in Michigan and California this week, Mr. Biden arrived in Phoenix on Wednesday night in advance of a speech at the Tempe Center for the Arts on Thursday morning, when he intends to honor the legacy of Mr. McCain, who represented Arizona in the House and Senate for 35 years before dying of brain cancer in 2018.The McCain project was compared by people familiar with the plan to a presidential-style library and museum for a man who tried twice to reach the White House but never did. In affiliation with Arizona State University, the new institution would house Mr. McCain’s papers as well as offer exhibits about his life, including possibly a reproduction of the so-called Hanoi Hilton, where he was held in North Vietnam as a prisoner of war for five and a half years.The announcement will be included in a speech that is meant to focus on what the president characterizes as a battle for American democracy as he faces the prospect of a rematch next year against Mr. Trump, who has been charged by both federal and Georgia state prosecutors with trying to subvert the 2020 election to hold on to power. In a summary that it distributed, the White House said defending democracy “continues to be the central cause of Joe Biden’s presidency.”The speech, according to the White House, will focus on the importance of American institutions in preserving democracy and the value of following the Constitution. It comes after three addresses Mr. Biden gave last year about the state of the country’s democracy and will brand Mr. Trump’s “Make America Great Again” movement a radical threat.“There is something dangerous happening in America,” Mr. Biden plans to say, according to advance excerpts released by the White House. “There is an extremist movement that does not share the basic beliefs of our democracy: the MAGA movement.”“Not every Republican — not even the majority of Republicans — adhere to the extremist MAGA ideology,” he plans to add. “I know because I’ve been able to work with Republicans my whole career. But there is no question that today’s Republican Party is driven and intimidated by MAGA extremists. Their extreme agenda, if carried out, would fundamentally alter the institutions of American democracy as we know it.”The renewed focus on Mr. Trump comes as Mr. Biden is being pressed to draw a sharper contrast with his once-and-possibly-future rival to remind Democrats and independents disenchanted with his own presidency of the stakes in next year’s election.Months of trying to claim credit for “Bidenomics,” as he calls his economic program, have not moved his approval numbers, as many voters, including most Democrats, tell pollsters that they worry about the 80-year-old president’s age. Democratic strategists argue that whatever Mr. Biden’s weaknesses, swing voters will come back to him once they focus on Mr. Trump as the alternative.In paying tribute to Mr. McCain, Mr. Biden hopes to reach out to anti-Trump Republicans and appeal to voters more generally in one of the battleground states that many analysts believe will determine the outcome next year. Mr. Biden and Mr. McCain served in the Senate together for many years and were friendly despite being from opposite parties. Even after running on opposing tickets in 2008, when Mr. McCain was the Republican presidential nominee and Mr. Biden was the Democratic vice-presidential nominee, they maintained a respectful relationship.Mr. McCain was one of the most vocal Republican critics of Mr. Trump, and Cindy McCain, the senator’s widow, endorsed Mr. Biden against the incumbent president of her party in 2020. In return, he appointed her to be his ambassador to United Nations agencies for food and agriculture in Rome. Earlier this year, she was appointed executive director of the United Nations World Food Program.Mrs. McCain will join Mr. Biden on Thursday morning along with other relatives of the senator, Gov. Katie Hobbs and members of Arizona’s congressional delegation. The president plans to use leftover money from the American Rescue Plan, the pandemic relief spending package approved shortly after he took office, to finance the new library.The library, described as a facility to provide education, work and health monitoring programs to underserved communities, will be formed in partnership with Arizona State and the McCain Institute, a public policy organization devoted to advancing issues like democracy, human rights, national security and human trafficking. More