After serving as ambassador from 2009 to 2013, Rice was tapped as national security adviser, a position she held until 2017.
She was considered as a possible vice presidential running mate this year for Biden and was floated as a possible secretary of state before Antony Blinken was ultimately nominated, The Washington Post reported.
McDonough's nomination as secretary of veterans affairs will require Senate confirmation.
He also has deep ties to the to Obama-Biden Administration, serving as White House chief of staff, deputy national security adviser and chief of staff of the National Security Council.
Biden on Thursday also officially announced his choice of Tom Vilsack, a former Iowa governor, to return as agriculture secretary; Ohio Rep. Marcia Fudge for housing and urban development secretary; and attorney Katherine Tai for U.S. trade representative.