Clubs are concerned they could breach contracts by furloughing players.
Three League One clubs, including Sunderland, have already placed their entire squads on the government scheme, while Millwall are among the Championship outfits considering doing the same.
The EFL wrote to clubs last month to indicate they can use the taxpayer-funded scheme for players but it remains unclear if employees can take holiday while furloughed.
The majority of players whose contracts are up in June are entitled to five weeks of paid holiday per year, three of which must be concurrent, leaving clubs fearing they may be forced to allow players to take paid annual leave in the coming weeks.
Given the unprecedented and ever-changing nature of the situation, the EFL is in regular contact with its 71 clubs and another update on furloughing could be issued on Thursday.
Clubs are also waiting for the outcome of talks between the EFL and PFA over wage cuts and deferrals before making a decision on players, with widespread hope that a collective agreement can be reached on a league-by-league basis.
The Premier League has not issued advice to its members on furloughing players, although CEO Richard Masters has defended clubs' right to place non-playing staff on the scheme.
It remains unlikely that top-flight players will be furloughed, with clubs now focussed on reaching unilateral agreements with their squads over wage cuts.
