Voters go to the polls in Arkansas, Maine, Nevada, North Dakota, South Carolina and Virginia.
The New York governor is positioning himself as a social liberal and fiscal centrist.
Rep. Tammy Duckworth (D-Ill.) is "Ready for Hillary," making her the second House lawmaker to back the group working to draft Hillary Clinton to run for president.
Sen. Elizabeth Warren (D-Mass.) does not have any interest in a bid for the White House, the freshman senator and former Obama adviser told The New York Times.
Warren said repeatedly in an interview with the paper that she did not intend to run for the Oval Office, and that sentiment was reportedly echoed by aides.
The declaration comes despite Warren's high popularity among the Democratic Party's liberal base, something that would undoubtedly buoy a presidential run. A Quinnipiac poll released last month that measured how warmly Americans felt toward potential 2016 candidates found Warren finishing third, behind only New Jersey Gov. Chris Christie and Secretary of State Hillary Clinton.
Party leaders were unable on Saturday to unify behind one of the 16 Democrats contesting the seat.
For the result to become official, it must first be certified by board commissioners.
Reports said Rangel was leading Democratic challenger state Sen. Adriano Espaillat by 994 votes as counting continued.
In a rally in Harlem, Rangel blasted his challenger for filing a new legal challenge to his seeming primary victory.
As counting continues, Rangel's reported margin of victory has shrunk from 6 points on Tuesday to 2.6, reports said.
Reyes was a former chairman of the House Intelligence Committee and a close ally of Minority Leader Pelosi.