The two German shepherds are the first pets to live at the executive mansion since the Obama administration.
Major burst onto the national scene late last year after Mr Biden, then president-elect, broke his right foot while playing with the dog at their home in Wilmington, Delaware.
The Bidens adopted Major in 2018 from the Delaware Humane Association, and Champ joined the family after the 2008 presidential election that made Joe Biden vice president.
The dogs moved into the White House on Sunday, following Mr Biden’s inauguration last week.
Michael LaRosa, spokesperson for first lady Jill Biden, said: “The first family wanted to get settled before bringing the dogs down to Washington from Delaware.
“Champ is enjoying his new dog bed by the fireplace and Major loved running around on the South Lawn.”
The dogs were heard barking outside near the Oval Office on Monday as Mr Biden signed an executive order lifting the previous administration’s ban on transgender people serving in the military.
Last week, the Delaware Humane Association cosponsored an “indoguration” virtual fundraiser to celebrate Major’s journey from shelter pup to first dog with more than 200,000 dollars (£146,000).
Major is the first shelter dog to ever live in the White House and “barking proof that every dog can live the American dream,” the association said.
Jennifer Pickens, author of a book about pets at the White House, said: “Pets have played an important role in the White House throughout the decades, not only by providing companionship to the presidents and their families, but also by humanising and softening their political images.”
Champ and Major follow in a long line of presidential pets, with Theodore Roosevelt having Skip, who is described by the White House Historical Association as a “short-legged Black and Tan mongrel terrier brought home from a Colorado bear hunt”.
Warren G. Harding had Laddie Boy, who sat in on meetings and had his own Cabinet chair and Franklin Delano Roosevelt had his beloved terrier Fala.
When he declared victory in the 2008 presidential race, Barack Obama told his daughters: “You have earned the new puppy that’s coming with us to the White House.”
Several months later, Bo joined the family, a gift from Senator Ted Kennedy.