However, he wants the former captain to change his coaching team.
Johnson, whose contract expires next month, has yet to let the RFU know if he wishes to remain in his job following England's disappointing World Cup campaign, which ended in a quarter-final exit at the hands of France.
Johnson, who skippered the Red Rose to Webb Ellis glory in 2003, has faced criticism for England's lumbering, conservative style of play, as well as a number of off-field incidents.
Those controversies led to the decision to fine Mike Tindall £25,000 and remove the centre from the international squad over events that took place on a night out in Queenstown, while James Haskell and Chris Ashton were handed suspended fines for their alleged conduct towards a female hotel worker in Dunedin.
But Greenwood, who scored 31 tries in his 55 England caps, believes despite the furore, the man he played with for club and country is deserving of another opportunity in the hot seat.
"The first thing is that Martin needs to decide if he wants to stay in the job and go through the process again," the 39-year-old, who this week announced he will don his boots again to take part in the Help For Heroes match at Twickenham on December 3.
"But if he does want to then I think he should keep his job. I played under Martin for many years and he is a true leader, an exceptional man. He is also a quick learner and I am sure he will have identified the things that went wrong during the World Cup and those mistakes will not happen again.
"Keeping Martin in place would not be a backward step. If you look at France, Marc Lievremont had a torrid time but they stuck with him and he got them to a World Cup final. Wales under Warren Gatland were in the middle of a very poor run a year ago and look what they did in New Zealand.
"It may be difficult for him but from the playing side of things I feel he could do with new faces in his coaching set-up. Sometimes players need to hear the message from fresh faces and with a fresh voice, to mix things up and get the best out of them."
Former Wasps head coach Shaun Edwards would appear to be a leading candidate should the likes of Mike Ford and Jon Wells be cast aside, having been linked with a role in the England set-up after his part-time contract with Wales expired at the end of the World Cup.
But while Greenwood concedes that the Wigan rugby league legend is a "brilliant option", the former three-quarter would also like to see younger coaches plying their trade in the Aviva Premiership rewarded.
"Shaun would obviously be a box-office appointment," said Greenwood.
"He has done some special things in his career, but you would have to say that he has not always been perfect. There have been periods where Wasps and Wales struggled.
"If you look around the Premiership there are other options. There are guys like Paul Gustard, who has done a great job on defence at Saracens, and Tony Diprose at Harlequins. Maybe they deserve the chance"
In addition to on-field disappointments, affairs within the RFU have been in a shambolic state since the departure of chief executive John Steele in June.
Steele's exit led to a period of bloodletting which claimed another victim last week when it was announced that acting chief executive Martyn Thomas was to step down from his role when his contract ends next month, as well as all other positions he holds within the game.
Greenwood feels the move is a step in the right direction, but says that Thomas' contribution to the game, including playing a key role in England winning the hosting rights for the 2015 World Cup, should be acknowledged.
"Martyn Thomas has been treated as the villain of the piece since Steele left." he said.
"But people need to remember that he has done some great things in his time with the RFU, particularly helping them to get the 2015 World Cup.
"Having said that, England are now effectively World Cup hosts and you need to have a fully-functioning RFU if that event is to be a success on and off the field.
"It's like when the 'Bloodgate' saga hit Harlequins. It happened and you have to deal with things and move on and this is a huge step forward for all involved."
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