It is not yet clear when Cole will leave the supermarket, although it is understood the timescale is over the next few months rather than weeks.
Cole will take over John’s responsibilities for media relations, internal communications, sustainability, government affairs and financial communications worldwide.
Bupa employs more than 70,000 people and has more than 22 million customers in 190 countries. It makes more than 70 per cent of its profits outside the UK.
Unlike the Sainsbury’s role, the Bupa role is represented on the executive board and Cole will report to chief executive Stuart Fletcher.
John is still in post at Bupa and the date of his departure for a global director of communications role at management consultancy McKinsey is yet to be confirmed.
Sainsbury’s marketing director Sarah Warby, in an internal announcement seen by PRWeek, said: "Alex has done a fantastic job for us over the past two-and-a-half years, promoting and protecting our brand in the external world.
"She has completely transformed the way that we engage with our stakeholders on the CR agenda, bringing the work we do on our 20x20 sustainability plan to life by finding innovative new ways to tell our story.
"And she has done an incredible job at helping us protect and enhance our reputation in the most challenging of times, most notably throughout the horsemeat issue. And of course, most recently she has managed the announcement of Justin [King]’s departure and Mike [Coupe]’s appointment seamlessly."
Cole joined Sainsbury's in December 2011 from Freuds, where she was MD of the agency's corporate arm for nine months. Before that she was global corporate affairs director at Cadbury.
She began her career in public affairs working as special adviser to the Rt Hon Jack Straw MP.
While Bupa hired Freuds for a UK consumer PR brief in 2010, a Bupa spokesman said the agency was no longer working for it. The company uses Brunswick for financial PR support.