Dan's Answer:There are some potential ways to do this. The most common is below
Ask your company if they allow for "withhold to cover" or "net exercise" transactions.
In this type of transaction the company keeps shares equal to the exercise cost and any tax withholding due at the time of exercise. They then deliver the net shares to you.
Some private companies don;t like doing this because it creates additional shareholders and the taxes you owe have to paid directly by the company. In a situation where the company is cash-poor, this may not be a great solution.