In addition to the answers that have already been given; the living seller can produce a death certificate to the Title company which will also help expedite the matter, in case one of the sellers does pass away.
But you did leave out some information that COULD tie things up, such as; do the sellers have a will or a trust in place, is the property that is being sold, free and clear or is there a mortgage, and is the property being sold possibly a short sale with the "selling" bank placing a "must close by date" on the deal.
All of these matters should be discussed with the seller's agent and a lawyer to try and take care of everything that needs to be addressed, PRIOR to one of the sellers passing. It sounds like time is of the essence in this matter so I would not waste any time in making sure BOTH sides are ready regardless of what happens with the seller's health.
This is my personal opinion only, not legal advice. I can not and do not give legal advice so you should seek legal counsel only from a licensed attorney in your state.
Perhaps the dying seller could provide power of attorney to someone else? Hopefully they are making the necessary provisions and arrangements that most do as life nears an end. Our contracts specifically address that the obligation to sell binds to the estate, heirs, etc. and thus there still may be an obligation to sell.
If you have an agent, have them address your concern with the listing agent. If you are working directly with the listing agent, discuss this with them.