Both awards were formally confirmed in October last year but the news was not published by the government and little publicised.
While the awards are approved by the Queen, they are not handed out by a member of the royal family but by government ministers.
It is understood that Ahrendts, an American who has helped build the British raincoat maker into an international fashion brand, will receive her award for services to business from a minister at the Department for Business Innovation and Skills (BIS).
Fellow American Gates will receive her award for services to philanthropy and international development from an official at the Department for International Development (Dfid).
Unlike awards for British nationals, which are announced in the Queen's birthday and New Year's honours lists, in June and January, foreign nationals can receive awards at any time during the year.
Ahrendts, the highest-paid chief executive in the FTSE 100 in 2012 with a pay package of almost £17m, receives her award just months before she leaves Burberry to become head of Apple's global retail division.
One of just a handful of women leading listed British companies, Ahrendts joined Burberry in 2006 where, alongside creative head Christopher Bailey, she has transformed the image of the brand.
The once fusty label favoured by elderly golfers is now feted as a leader in digital communications, live-streaming catwalk shows around the world and arming every shop assistant with an iPad.
The Gates Foundation supports research and development projects related to tackling poverty, health issues and education.
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