Dyke wants to increase the minimum number of home-grown players in club squads from eight to 12, however he is facing opposition from the Premier League.
The proposals also include changing the rules so that 'home-grown' means having trained in England for three years before the age of 18 rather than before 21.
Graham Taylor, Glenn Hoddle, Kevin Keegan, Sven-Goran Eriksson and Steve McClaren have signed a letter addressed to Dyke, the Daily Telegraph reports.
According to the newspaper, the five former Three Lions bosses warn the English game will suffer if it blocks the proposals from Dyke and the FA's England Commission.
"We urge everyone in the English game to get behind them as quickly as possible," the letter reportedly reads.
It adds: "Failure to do so risks England falling further behind the leading football nations and will only make it harder to end the long wait to win the World Cup."
Only 81 of the 220 Premier League starters in the last round of matches were eligible for England selection - 36.8 per cent.
The managers add: "There are many reasons why England has failed to win any major honours since 1966 and each of us bears a portion of that responsibility.
"However, as the England Commission's evidence has demonstrated, the pool of English talent playing at the very top level is shrinking and it's an undeniable fact that this is a clear disadvantage for any England manager."
A draw for England in Italy was a respectable result, and Roy Hodgson's team has encountered no trouble in its EURO 2016 qualifying group to date, winning all five of its matches.
There are signs of encouragement with the emergence of a new generation of young players, the former England bosses reason.
"With promising players such as (Ross) Barkley, (Raheem) Sterling and (Harry) Kane starting to break through, it is clear we can still produce top young talent. The one thing they are not being offered is the most important of all: opportunity."
