WASHINGTON—Sen. Elizabeth Warren said that Sen. Bernie Sanders, one of her rivals for the Democratic presidential nomination, told her during a private 2018 meeting that a woman couldn’t win the U.S. presidency—a statement Mr. Sanders denies.

Ms. Warren, in a statement distributed by her campaign Monday evening, said that she and Mr. Sanders met for more than two hours just over a year ago when each senator was laying the groundwork for a White House bid.

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