Beto O’Rourke, the Texas Democrat whose near miss Senate bid last year catapulted him to national fame, said on Tuesday afternoon that he would decide by the end of February whether to run for president in 2020.

Another potential 2020 candidate, Senator Amy Klobuchar of Minnesota, disclosed the timing of her own decision after President Trump’s State of the Union address. Ms. Klobuchar said she would reveal her plans about 2020 on Sunday at an event in Minneapolis.

If she runs, Ms. Klobuchar would be the fifth woman currently serving in Congress to enter the Democratic race, which is already the most diverse in history. She and Mr. O’Rourke are among several Democrats who are now eyeing the race, including Senator Bernie Sanders, former Vice President Joseph R. Biden Jr. and Michael Bloomberg, the former mayor of New York City.

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Mr. O’Rourke, in an interview in Times Square with Oprah Winfrey, said family considerations were the only obstacle to a campaign for the Democratic nomination.