Players and staff are due to be paid on Friday but the club's owners, Gulf Finance House Capital (GFH), need £2m.
Cellino is attempting to buy the club but the Football League blocked his proposed takeover on Monday.
Senior figures at Elland Road insist the club will not go into administration.
Italian Cellino, who agreed to buy a 75% stake in the club in February, has paid the club's wage bill for the past two months, while managing director David Haigh has put in around £1.5m of his own money to keep the club running.
According to reports in Yorkshire, It is understood the Championship club are losing between £1m and £1.5m per month.
Though Cellino is set to appeal against the decision to block his takeover, he is understood to be unwilling to add to the £6m-plus he claims he has already invested.
However, GFH believe they have an agreement with Cellino's company, Eleonora Sport, that commits them to cover the club's costs for a six-month period, even if their takeover fails.
If Cellino continues to hold off investing more funds, GFH may be forced to borrow the money.
Leeds were beaten 4-1 by Bournemouth on Tuesday, leaving manager Brian McDermott to say the club's performances were being affected by off-field issues.