Dr Mike Williams, non-executive director of Derriford Hospital in Plymouth, said it was "reliant on forward payments" from commissioners.
Plymouth Hospital NHS trust papers show the cash balance was £0.2m at the end of August, £6.7m lower than July and £3.8m lower than planned.
The hospital said it would not run out of cash this financial year.
A spokesman said payments from the Trust Development Authority and health commissioners Northern, Eastern and Western Devon Clinical Commissioning Group (CCG) would sustain the hospital's finances.
Dr Williams' personal views, published in a letter to The Guardian newspaper, said the hospital was also "running out of the precious commodity of staff in key specialities".
The trust papers report a £2.56m shortfall in performance payments due to a shortage of beds and surgical staff.
The trust also incurred greater than planned fines for A&E targets, cancer and cancelled operations.
Ann James, chief executive of Plymouth Hospitals NHS Trust, said: "We have a plan in place to manage our finances and particularly our cash position until the end of the financial year.
"This plan includes support from our commissioners in the shape of some temporary advances, which will be returned during the course of this financial year.
"In the face of these challenges, it is particularly reassuring for patients to know that independent inspectors rated the care our staff give as "outstanding" when they visited us this year."
A Department of Health spokesman did not comment on the situation at Derriford but said: "We believe in the values of the NHS which is why we have invested the £10bn the NHS has said it needs to fund its own plan for its future.
"There is no denial that the environment is tough but the NHS must deliver its side of the plan with cost-controlling measures like clamping down on rip-off staffing agencies and expensive management consultants."
A director of Devon's biggest hospital has warned it is "fast running out of cash".