Leeds and Burnley have written to the Premier League claiming that Everton have seriously breached financial rules.
The two clubs sent a joint letter to the League on 13 May asking for an independent commission be set up to investigate the matter within six weeks of their correspondence.
Leeds and Burnley have threatened to bring a claim for substantial damages against the Premier League and Everton after the Merseyside club recorded losses of £371.8 million over the last three years.
The Premier League's profit and financial sustainability rules allow clubs to lose a maximum £105m over a three-year period or face sanctions, which include points deductions for serious breaches.
The PA news agency understands that Leeds and Burnley view the issue as a matter of sporting integrity, asking the Premier League to enforce its own rules - and a potential points deduction - if Everton are found guilty of breaching financial regulations. The Toffees say they have complied with the rules.
Leeds and Burnley, one of whom will be relegated on the final day of the season on Sunday, had been involved in a three-way relegation battle with Everton.
But Frank Lampard's side secured their top-flight safety on Thursday by beating Crystal Palace 3-2 at Goodison Park.
The Premier League declined to comment when contacted by the PA news agency, as did Leeds and Burnley.
Losses caused by the Covid pandemic can be written off, but Leeds and Burnley have raised concerns over Everton's losses.
Everton's pandemic-related losses were more than three times that of other similarly-sized clubs.
The Merseyside club announced their annual results in March and said £170m of the losses were related to the "dramatic and ongoing financial impact" of the Covid pandemic.
Everton were one of the biggest spenders in the last January transfer window, with a reported #30m spent on Vitaliy Mykolenko and Nathan Patterson.
Dele Alli arrived from Tottenham and loan signing Donny Van de Beek was also brought in on big wages from Manchester United.
An Everton spokesperson told the Times: "We have worked so closely with the Premier League to make sure we are compliant we are comfortable we have complied with the rules.
"External auditors have told us what we can and cannot claim against the pandemic.
"If they want to take legal action then they can do so by all means."
Leeds and Burnley are also understood to have expressed their concerns that the league has not acted appropriately or quickly enough.
Those concerns could now lead to potentially the biggest legal challenge the Premier League and a club have faced for 15 years.
West Ham paid £20m to Sheffield United in 2007 after being fined for breaching third-party rules in signing Argentina forward Carlos Tevez.