Staff at the club, who dropped out of the Football League for the first time in 112 years on Saturday, are still waiting for their March wages 24 days after they were due to be paid.
Owner Francesco Bechetti has had no communication with Orient staff over their missing payments.
Players and staff released a statement last week seeking assurances they will be paid this month and urged the EFL and Football Association to take immediate action after the club's failure to respond.
Orient survived a winding-up order last month over unpaid taxes, reported to be £250,000, with Becchetti given until June 12 by the High Court to pay off debts or sell up.
The EFL released a statement on Monday evening saying it was frustrated about the events at Brisbane Road and the governing body has warned the club's directors that "silence is not an acceptable response".
It read: "The EFL remains exceptionally concerned about the current situation at Leyton Orient.
"We are also aware of the frustration being expressed by the club's staff. This is shared by the EFL executive, as we are similarly seeking answers to a number of important questions from the directors of the club. Silence is not an acceptable response.
"The EFL last met with Leyton Orient chief executive Alessandro Angelieri on March 9. At this meeting, we received confirmation that funds would be made available to dismiss the winding up petition and fund the club going forward.
"The EFL board received an update on the situation at its meeting on April 6 with the club being asked to provide a further commitment following widespread reporting that the required funding had not been made available. No response has been received to date.
"The EFL executive are continuing to work with the club's dedicated staff to ensure that appropriate arrangements are in place for Saturday's final home game of the season and we would like to place on record our thanks for those that are making this possible, including the local safety advisory group. Additionally, a meeting has been arranged with Leyton Orient Fans' Trust, to try and keep those parties with the club's best interests at heart, united.
"While Leyton Orient may have been relegated to the National League, the EFL will continue to do everything it can to ensure that the future of the club is secured."
The EFL also defended its 'fit and proper person' guidelines that all football club owners have to pass.
The statement said: "There has been significant comment in recent days about the EFL's owners' & directors' test in the context of the ownership of Leyton Orient. It is important that supporters understand that the test governs the eligibility of who is able to own a club; it does not also ensure that those individuals have the capability to manage it properly.
"The role of the EFL is one of a competition organiser and the management of clubs is a matter for their directors, as long as their conduct is not in breach of any of our regulations. There are any number of actions the EFL might wish to take, but it can only ever act in accordance with those regulations."
Earlier on Monday, Barry Hearn, who spent 19 years as O's chairman, quit as the club's honorary life president in protest at the failure to pay staff.