Mr. O’Rourke made the decision to quit the race in the middle of this week, on the eve of a gathering Friday of Democratic presidential candidates in Iowa, according to people familiar with his thinking. He is not expected to run for any other office in 2020, despite persistent efforts by party leaders and political donors to coax him into another bid for the Senate.

His campaign has been under extreme financial strain, and Mr. O’Rourke’s advisers concluded that proceeding in the race might have meant making deep cuts to his staff in order to pay for advertising and other measures to compete in the early primary and caucus state.