In theory yes. Obviously proving this sort of scenario is rather impossible but biologists have debated this area extensively. Although the blood supply has stopped there will still be oxygenated blood within the brain, and as we already know the brain can continue for about 3-4 minutes without fresh supplies - hence using CPR can revive a person even after the heart has stopped pumping for a minute or two. The question is - what will you be conscious of? The brain will shut down and only perform reflex action tasks (attempting to breathe, but no lungs, attempting to send signals to the heart, signal has been cut) so you won't see, hear, smell, touch, taste, in my opinion, only be aware of thoughts going on in your mind. In summary i don't think you will get a definitive answer here, just opinion
As Andy points out there are some functions of the body that continue after death, which misleads people into thinking the body is still alive; dead bodies have been known to breathe out, or even fart, long after they went to the morgue!