WASHINGTON — As Donald J. Trump prepared to take his beauty pageant to Moscow in 2013, he relished the participation of one man above all: President Vladimir V. Putin. “Do you think Putin will be going to The Miss Universe Pageant in November in Moscow — if so, will he become my new best friend?” he wrote on Twitter.

Mr. Putin did not attend — in fact, Mr. Trump has yet to meet him, by all accounts — but his admiration for the Russian leader, shared by some of his campaign advisers, is clear. It stems from an affinity for brash, swaggering politics and business dealings in Russia that has overturned decades of Republican orthodoxy about Russia and its leader as an aggressive authoritarian whose values are inimical to those of the United States.

Hillary Clinton, by contrast, has a far more contentious history with Mr. Putin, despite the “reset” President Obama’s administration pursued with Russia during her tenure as secretary of state. In 2011, Mr. Putin accused her of instigating the huge protests that erupted after parliamentary elections marred by fraud.

“She set the tone for some actors in our country and gave them a signal,” he said darkly, going on to accuse her of engaging in “active work,” an old term of art for covert K.G.B. operations. When she compared Russia’s intervention in Ukraine to Hitler’s moves in the 1930s, he said she had “never been too graceful with her statements.”