Cameron will cede power to 59-year-old May after visiting Buckingham Palace to resign on Wednesday. May will become the UK's second female Prime Minister after Margaret Thatcher.
In one of the final tasks of his premiership, Cameron on Tuesday gathered his top team for a final time.
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Cameron had announced he would be stepping down for a new prime minister to take the Brexit negotiations forward with the European Union a day after the June 23 vote in favour of Britain leaving the economic bloc.
"It was emotional but quite rightly we spent a lot of time on the Government business. But then we had a period in which people paid tribute to the extraordinary service the Prime Minister has given, led off by George Osborne and Theresa May our new prime minister. And then we looked forward to getting back to work," the leading Brexit campaigner told Sky News.
Whittingdale said his Cabinet role was "entirely a matter" for Theresa May to decide on but said he was "very happy to continue" in Government if asked to do so.
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