The 65-year-old American, who is leaving the MLS club to take up the national post for the second time, led his country to the 2002 World Cup quarter-finals in his first spell in charge.
Arena, who initially managed the USA between 1998 and 2006, replaces former Germany international Klinsmann, who was sacked by the United States Soccer Federation on Monday after back-to-back defeats to Mexico and Costa Rica in World Cup qualifying.
"Any time you get the opportunity to coach the national team, it's an honour," Arena is quoted by BBC Sport as saying.
"I'm confident that we'll take the right steps forward to qualify for the 2018 World Cup in Russia."
Arena's first day back in charge of the US will be on December 1, and his first match will be a qualifier at home to Honduras on March 24 next year.