The former Manchester United and England legend, who is no longer under contract after it ran out on May 9, has been considering whether to stay or go for weeks.
Now the club want him to make up his mind before the start of next week. But the current financial crisis could force his hand and spell his departure.
Robson is prepared to take up the hotseat at Valley Parade when the new campaign starts in August if City are not in administration.
That's unlikely though after a proposed meeting to thrash out a repayment plan with creditors under a Company Voluntary Arrangement was shelved last month.