Johnson, who was captain when England lifted the Webb Ellis trophy in 2003, has been installed as one of the early favourites to take over from head coach Andy Robinson if, as expected, he leaves the post on Tuesday.
But Dawson, who retired at the end of last season, does not believe his former team-mate will be in charge when England begin their RBS 6 Nations campaign against Scotland at Twickenham on February 3.
"I don't think Martin Johnson would take it on now because I don't think there would be a role which would be suitable for him," Dawson said.
"But they desperately need to have some kind of figurehead who has a pure understanding of international rugby."
Though Dawson does not advocate the sacking of Robinson's coaching staff, which was revamped earlier this year, he is adamant there should be greater involvement for the players who tasted triumph in Sydney three years ago.
And he is scathing of the current regime's lack of playing experience at the highest level.
"Let's get some involvement of those players - Martin Johnson, of course, Jason Leonard, Paul Grayson, Will Greenwood, Neil Back, Lawrence Dallaglio and Mike Catt," he told BBC Radio Five.
"Get these guys involved because they know how to win Test matches.
"Look at the coaching staff we have at the moment. Is it eight or seven caps they have between them? That's just not enough.
"They don't understand the game of international rugby, so how can they coach it?"
However, the 34-year-old former Northampton and Wasps player is not convinced potential candidates will be queueing up to take on what promises to be one of the toughest jobs in world rugby.
And he believes Rugby Football Union chiefs must be realistic in their goals for next year's World Cup and instead focus on New Zealand in 2011, when any structural changes will have had time to take effect.
"It's not a challenge people are going to be grasping for particularly quickly," Dawson said.
"However, it depends how the RFU look at it. Do they turn round and say 'realistically, how are England going to do in this World Cup'?
"They would be unbelievably fortunate to get to the semi-finals with the state of play at the moment.
"So, do they totally reconstruct everything from now on? It can't get any worse, surely, than it is at the moment.
"Potentially we can start building for 2011, because let's be realistic, England need to get themselves out of the mire to get anywhere near qualifying for the semi-finals, let alone the final."