The Smithsonian plans to pay for both works with $500,000 in private donations of which $300,000 has been raised so far. This amount covers everything including each artist’s fees, which Ms. Sajet would not specify. Ms. Sherald’s paintings typically sell for $15,000 to $25,000.

There is no prescribed format for an official portrait, Ms. Sajet said, except that it be a painting — not a work on paper — and that the artists “be respectful of both the person and the position they hold.”

Official portraits have ranged from “highly formal to much more relaxed,” Ms. Sajet added, citing as examples Ron Sherr’s 1994 portrait of George H. W. Bush standing in a suit and tie at the White House and Robert Anderson’s 2008 likeness of George W. Bush in an open-collared shirt on a couch at Camp David.

The selection of Ms. Sherald, who typically depicts African-Americans doing everyday things — two women in bathing suits, a man holding a child — has historical significance. “It’s as if she’s saying, ‘Let’s be clear: the President and I are African Americans and proudly so,” Mr. Staiti said, “and these portraits are going to have an African-American vibe — they’re going to break out of that rather staid tradition. I think it’s important and I think it’s political,” he added, referring to Mrs. Obama’s choice.