If confirmed by the Senate, Young would be the first Black woman to lead the Office of Management and Budget while Coloretti, who is Filipino American, would become one of the highest-ranking Asian Americans in the Biden administration.
It would be the first time that two women of color will lead the agency, which helps the president develop an annual budget and monitors how tax dollars are spent.
"Today it's my honor to nominate two extraordinary, history-making women to lead the Office of Management and Budget," Biden said in a video announcement released Wednesday while he spends the Thanksgiving holiday on Nantucket island in Massachusetts.
Young has served as acting director of the office for much of the year. Biden's first nominee for the director's position, Neera Tanden, faced bipartisan criticism for her previous attacks on lawmakers, and she ultimately withdrew from consideration.
Biden said Young has "continued to impress me" during her time as acting director.
She faces her own Senate confirmation vote, but she was confirmed to be deputy director in March with nearly two-thirds support and the backing of multiple Republicans. As the former staff director of the House Appropriations Committee, Young had the support of top Democratic leaders, including Speaker Nancy Pelosi.
Pelosi, in a statement issued Wednesday, said Young's nomination is "well-deserved."
Coloretti would join the administration from the Urban Institute think tank, where she is a senior vice president.
Biden said his nominees are "two of the most experienced, qualified people to lead" the budget office, and he called on the Senate to vote quickly to confirm them for the posts.
The head of the Office of Management and Budget is tasked with putting together the administration's budget and overseeing a wide range of logistical and regulatory issues across the federal government.
Jaffe reported from Washington.
This story has been corrected to say that the Senate confirmed Young to be deputy director, not acting director.