WASHINGTON—Massachusetts Sen. Elizabeth Warren on Monday announced the formation of an exploratory committee for president, making her the first top-tier candidate to make a formal move toward running for the 2020 Democratic nomination.

Ms. Warren, long among the highest-profile members of the Democratic Party’s liberal wing, is one of at least two dozen candidates who might seek the party’s presidential nomination. Her political stature within the party has diminished since her October release of a genetic analysis that she said provided evidence of her Native American ancestry. That was received poorly by some Democrats, and President Trump mocked her.

“Today, I’m launching an exploratory committee for president because the outcome of this election will depend on you,” Ms. Warren says in a four-minute video that includes images of President Trump, Treasury Secretary Steven Mnuchin and departing GOP House Speaker Paul Ryan. “For the last two years, millions of people have done more than they ever thought they would to protect the promise of America. And here’s what we learned: If we organize together, if we fight together, if we persist together, we can win. We can and we will.”

A federal exploratory committee will allow Ms. Warren to travel the country and raise money from supporters before launching a formal campaign. Exploratory committees let federal candidates raise and spend money to “test the waters” for a campaign without transparency requirements mandated by the Federal Election Commission once a candidate formally declares they are seeking office.

Several other Democrats, including as many as eight senators, are expected to launch campaigns or exploratory committees in January to begin raising money before the first major fundraising deadline on March 31.