Following a disastrous 16-0 points whitewash in the Women's Ashes earlier this year, England is searching for both a new head coach and captain after Jon Lewis and Heather Knight were removed from their positions.
While several names may be in contention to replace Lewis, Greenway insists that now is the ideal moment to appoint Edwards, who has enjoyed significant success as a coach.
"I played most of my career alongside her and have coached with her as well. I truly believe she is the only name England should be looking at," said Greenway, 39, speaking to BBC Test Match Special.
"I haven't spoken to her, but I think this is the perfect time for her to step into this role. When the job was initially available, when Lewis took over, she probably wanted more experience in a head coach role, and now she's gained that."
Greenway acknowledged that the England and Wales Cricket Board (ECB) will go through their selection process, with potential candidates such as former Australia head coach Matthew Mott and England A coach Jon Lewis possibly in the mix. However, she insists England should look no further than Edwards.
Edwards enjoyed a 20-year international career before retiring in 2016, spending a decade as England's captain. Her successor, Heather Knight, led the team to World Cup glory in 2017.
Since transitioning into coaching, Edwards has won multiple titles, including two Women's Premier League (WPL) championships with Mumbai Indians, The Hundred with Southern Brave, and five domestic trophies with Southern Vipers. She has also been appointed as Hampshire's head coach for the revamped county structure in England and Wales in 2025.
After England's Ashes humiliation, Edwards expressed her willingness to help rebuild the team on the BBC Stumped podcast. Greenway believes her coaching style would be instrumental in England's progression.
"In women's cricket, we've seen different coaching styles—some too restrictive, others too relaxed. Jon was quite lenient, allowing players a lot of freedom, but Edwards has the right balance," Greenway explained.
"She holds players accountable on and off the field but also brings fun to the team. She understands that players aren't robots and knows when to give them breathing space."
World Cup-winning spinner Alex Hartley also endorsed Edwards, describing her as a "brilliant, outstanding coach" who can raise the team's standards and help them perform under pressure.
"She lets players figure things out on their own but doesn't tolerate nonsense, and that's exactly what England need right now," Hartley added.
While Edwards is the frontrunner for head coach, the choice of England's next captain remains uncertain. Nat Sciver-Brunt, Knight's deputy, appears to be the leading contender, but Greenway and Hartley also highlighted off-spinner Charlie Dean as a potential future leader.
"A major issue over the past five or six years has been the lack of a clear successor for captaincy," Hartley said.
"They need a coach who can set the right culture and fix issues within the squad, allowing the new captain to come in with strong ideas.
"I'd love to see a young, fresh leader like Charlie Dean take on the role. She has the potential to be an exceptional captain for years to come."
