The chat between Mr Johnson and one of the world's richest couples comes ahead of of the Global Vaccine Summit, which is being hosted by the UK on June 4.
The summit hopes to raise at least £6bn for a vaccine after £6.5bn was raised in a global funding drive co-hosted by Boris Johnson in early May.
The UK and The Gates Foundation - the couple's philanthropic organisation - are among the world's biggest contributors to international attempts to find a vaccine for Covid-19.
In late March, Mr Gates thanked Mr Johnson and the UK for a "vital commitment" of £210m toward vaccine efforts.
In May, the Gates Foundation said it would spend $245 million to fight the coronavirus pandemic after already pledging over £100 million in March.
Bill and and Melinda Gates are said to be one of the world's richest couples. with a net worth estimated to be at least £81.5 billion.