"The Club would like to thank both for all their efforts throughout their short time as members of the Board," Villa said in a statement.
"A search to appoint a new manager remains a priority. The Chairman, Steve Hollis, is currently engaged in extensive negotiations in relation to the sale of the Club."
King, a life-long Villa supporter who led the Bank of England through one of the country's biggest-ever economic downturns during his 10 years in office from 2003 to 2013, joined the board in February.
Bernstein arrived as part of a boardroom shakeup in March, with responsibilities for reviewing the "football side" at Villa.
"Unfortunately it had proved impossible to implement the agreed structure and my position has quickly become untenable," the BBC quoted him as saying in a resignation letter.
"The issues at the club are fundamental and the solutions are radical and do not lend themselves to compromise."
Villa, who have won only three matches in the league this season, were relegated last Saturday after losing 1-0 to Manchester United. American owner Randy Lerner has been trying to sell the club since last May.
The former European champions, and seven-times champions of England, last played outside the top flight in 1988.
(Reporting by Alan Baldwin, editing by Gareth Jones)
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