Only time will tell whether he gets his wish.
No-one should get too carried away, though.
Britain has had only a handful of non-white cabinet ministers. Indeed, Mr Cameron's new cabinet, following his election victory in May, has just two minority-ethnic cabinet ministers.
It is all a far cry from 1987, when four new black and minority-ethnic MPs - all Labour - entered the House, a symbolic and powerful moment for many observers.
Paul Boateng had the most successful ministerial career, eventually making it into the cabinet as chief secretary to the Treasury.
Much of the commentary focused on his being the first black cabinet minister and some speculated that he might rise to the very top. It was not to be. He now sits in the Lords.
Keith Vaz rose to become the first Asian minister in the Commons in 1999.
Mr Vaz's newsworthy quotes keep him in the public eye as the long-serving chairman of the home affairs select committee.
Diane Abbott has a big political and media profile but has stayed on the back benches, as did Bernie Grant, before his premature death.
The Tories at that time had no real equivalent representation in terms of black and minority-ethnic MPs.
There was a lingering suspicion that the majority of voters from these communities didn't like the Conservatives.
There was a damaging public row when a black candidate, John Taylor, was selected to fight Cheltenham in 1992. There was opposition from the local association and the campaign itself was marred by racist jibes. Taylor lost.
It was the Blair-Brown years that changed the landscape at ministerial level.
The Tottenham MP, David Lammy, was one early star. He rose quickly and, having been elected at just 27, he was immediately weighed down by the "Britain's next black prime minister" label.
He didn't last but he has reinvented himself as a vocal backbencher and is in the mix to be Labour's choice for London mayor.
Baroness Scotland, the first ethnic minority attorney general, was another prominent appointment. Then there was Shahid Malik, who became Britain's first Muslim minister in 2007.
When David Cameron came to power in 2010, he brought in Baroness Sayeeda Warsi. She was the first Muslim woman cabinet minister.
Her star burned brightly for a short time but she is no longer in government. Instead, she has carved a role as a critical voice, particularly on relations with the Muslim community.
Another casualty was Shailesh Vara, who lost his post as a government whip. He is currently parliamentary under-secretary of state at the Ministry of Justice. and remains influential in Tory attempts to appeal to the British Asian community.
The Liberal Democrats, now a crushed force in Westminster, did not have a single non-white MP in the 2010-15 Parliament.
The twists and turns of political life make predictions difficult.  The  Business Secretary Sajid Javid is seen by some as a future leader. His ethnicity - the son of an immigrant from Pakistan - is of less interest than his political views, perhaps another sign of a gradual shift in perceptions.
Priti Patel in the employment portfolio is another rising star - she became the first female minister of Indian descent last month.
Javid and Patel would once have been that rare breed - minority-ethnic Westminster politicians with right-wing views. No longer. Patel's beliefs on welfare reform attract more interest than her ethnicity.
On the Labour side, Chuka Ummuna was seen by many as the leading candidate to be the new Labour leader - until his withdrawal from the race. He entered Parliament just five years ago but is already a senior Labour figure and a regular on TV.
Sadiq Khan is another example of a big and influential Westminster politician, experienced in shadowing ministers, and now hoping to run for London mayor.
The new House of Commons has a record number of non-white MPs, who make up about 6% of the new Parliament, up from just over 4% in 2010.
Critics say this is encouraging but goes nowhere near far enough if Parliament is to truly represent all of the population.
David Cameron said in a speech last November that he wanted to see an ethnic minority prime minister to prove "Britons of all backgrounds can achieve".