It would seem by the phrase, "whomsoever believes in Him" would mean anyone and everyone. But as with the word "world", we must define whoever. To do this correctly, we need to know some basic facts.
In the Bible, people who are not saved are called:
in the flesh
People who are saved are called:
in the Spirit
The clear teaching of the Bible is that of flesh and spirit, life and death, heaven and hell. So now the question remains, who does the "whosoever" refer to out of these two groups?
But the natural man does not receive the things of the Spirit of God, for they are foolishness to him; nor can he know them, because they are spiritually discerned.(1 Cor. 2:14)
Because the carnal mind is enmity against God; for it is not subject to the law of God, nor indeed can be. So then, those who are in the flesh cannot please God.(Rom. 8:7-8)
Jesus answered and said unto him, "Most assuredly, I say unto you, unless one is born again he cannot see the kingdom of God.." (John 3:3)
By the Spirit moving upon us, raising us from the spiritual dead-only then can we have the ability to believe. We become the "whoever" only after God has done His work-it's not the other way around. The natural man cannot be the "whoever" for he is in the flesh and unable to believe.