But both men have since decided not to apply for a position with the RFU, although Mallett admits he was approached by England.
"I was interested to hear what the RFU had to say," said ex-Italy coach Mallett, who spoke to the RFU before Johnson's resignation.
"After mulling it over for a few days and discussing the opportunity with friends and family, I have decided that I will not be making myself available.
"I am happy to have returned home to South Africa after a tremendous four-year term with the Italian national team. We are settled in Cape Town and it is my wish to be able to enjoy time with my wife and kids."
Like Mallett, White would be a marquee signing for the RFU who are currently awaiting the results of two separate reviews into England's disappointing quarter-final exit at the recent World Cup.
And while Mallett guided the Springboks to 17 consecutive Test victories in 1997 and '98, White won the World Cup in 2007 and would arguably have been an even bigger coup for English Rugby.
But the 48-year-old IRB Hall of Fame inductee is only part-way through a long-term contract with Australian Super 15 side the Brumbies and has no intention of jumping ship, even for a return to the international arena.
"Jake has a contract for another four years with the ACT Brumbies," his representative Craig Livingstone told BBC Sport.
"He won't be available for the England job. It's just bad timing."