Burnley and Leeds want Everton's finances thoroughly investigated and have asked for an independent commission to be set up to decide whether Everton have broken Premier League rules and whether they have a case to answer.
They have asked for the commission to be set up within six weeks of the date of their letter (May 13).
If there has been a breach they want rules to be enforced, and the maximum punishment is a points deduction.
They want this dealt with before next season's fixtures are released on June 16. They want a quick resolution - unlike the Premier League's investigation into Manchester City's alleged breaches of financial fair play rules which have been going on for more than three years.
There is believed to have been encouragement from other Premier League clubs to take this course of action.