In a statement, the Biden campaign said Trump "publicly and repeatedly invited, emboldened, and even tried to coerce foreign interference in American elections".
Evanina said Andriy Derkach, a pro-Russia Ukrainian politician, has been "spreading claims about corruption - including through publicized leaked phone calls" to undermine Biden's campaign and the Democratic Party.
Trump supporters in the US Senate have launched investigations questioning Biden's son Hunter's involvement in alleged business activities in Ukraine.
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Evanina said "Kremlin-linked actors" also are trying to "boost President Trump's candidacy via social media and Russian television".
He said his agency assessed that China would prefer that Trump not win re-election because Beijing regards him as too unpredictable.
He said China has been expanding efforts to influence US politics ahead of the election to try to shape US policy, exert pressure on US politicians it regards as anti-China, and deflect criticism of China.
Evanina said Iran is likely to use online tactics such as spreading disinformation to discredit US institutions and President Trump and to stir up US voters' discontent.
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The leaders of the Senate Intelligence Committee, Republican Marco Rubio and Democrat Mark Warner, thanked Evanina for his warning in a statement.
They added that all Americans "should endeavor to prevent outside actors from being able to interfere in our elections, influence our politics, and undermine confidence in our democratic institutions".
Tim Murtaugh, a spokesman for Trump's re-election campaign, insisted that Trump has been "tougher on Russia than any administration in history".
"We don’t need or want foreign interference, and President Trump will beat Joe Biden fair and square," Murtaugh added.
Many officials who oversee US election technology and outside security experts now worry less about hacking in the elections than about misinformation and logistics such as a shortage of poll workers and slowdowns at the US postal service.
(Writers: Mark Hosenball, Jeff Mason, Trevor Hunnicutt | Editors: Scott Malone, David Gregorio, Sonya Hepinstall)
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