The letter, written and circulated by the Booker campaign, was sent to Mr. Perez and his top deputies Saturday afternoon. It comes amid angst that Thursday’s debate in Los Angeles will include just one candidate of color — the businessman Andrew Yang — among the seven qualifying participants.

Mr. Booker did not qualify for next week’s debate and is unlikely to meet the criteria for the debates in January and February unless thresholds are lowered or he sees a significant improvement in his polling numbers.

The nine candidates — former Vice President Joseph R. Biden Jr., Senators Bernie Sanders of Vermont, Elizabeth Warren of Massachusetts and Amy Klobuchar of Minnesota, Mayor Pete Buttigieg of South Bend, Ind., the businessman Tom Steyer, the former Housing secretary Julián Castro, Mr. Yang and Mr. Booker — asked Mr. Perez to use either a polling or fund-raising threshold, but not both.