Buyouts are usually for more than the ORIGINAL value of a stock. That's because the price "premium" represents an incentive for holders to "tender" their shares to the would-be buyer.
Sometimes in these situations, the stock price rises above the proposed buyout price, in anticipation of a higher takeover bid from a SECOND party (that may or may not materialize).
To answer the other part of the question, does a bidder have a chance of taking over a dying company for less than the market price? That is a strategy sometimes referred to as a "take under," and it has not been a notably successful strategy.
That's because it goes against "human nature" (of the seller). "Where there is life, there is hope." They would seldom accept a lower price for "sure" survival, when the market is telling them that they are worth a higher price. Very few people realize that the market may disappear tomorrow. Think of all the homeowners who won't cut their price, but insist on bids that meet recent "comps."
And if the company is really dying, the prospective buyer may be best served by waiting until it does, and then pick up the individual pieces at auction.