A Wizards forum thread had an interesting question that was left unanswered. If I have only two black mana available, can I cast Fatal Lore if my opponent has a Spellwild Ouphe on the battlefield? The issue is it is a legal play only if my opponent chooses the destroy creatures option (since I could target the Ouphe, making it cost two lest to cast).
Since it is a legal play under some outcomes of the process of casting it, it is a legal play. If your opponent chooses the mode where you draw cards, it will become an illegal spell and the game will be reverted to the state it was in before you began casting it.
The precedent for this comes from a card specific ruling on Selvala, Explorer Returned:
If you activate Selvala’s ability while casting a spell, and you discover you can't produce enough mana to pay that spell’s costs, the spell is reversed. The spell returns to whatever zone you were casting it from. You may reverse other mana abilities you activated while casting the spell, but Selvala’s ability can't be reversed. Whatever mana that ability produced will be in your mana pool and each player will have drawn a card.
From this, we can tell that you are allowed to cast spells you might be able to pay for.
As Hackworth and various members on the forum point out, the rules do not explicitly prevent you from attempting to cast spells you have no way of paying for in any circumstance, but doing so doesn't ever help you. At best, it reveals a card in your hand with no other consequence. At worst, it could get you disqualified in a tournament for repeat offenses of slow play or cheating (if you are attempting to cast an illegal spell to make an opponent reveal information in their reaction).
You should be able to cast it. When you declare that you want to cast it, you don't yet know whether it will be possible to pay, but it could be possible, depending on your opponent's choice. If your opponent chooses that you draw 3 cards, you won't be able to continue casting it. But if your opponent chooses the destruction mode, then you will be able to finish casting it.
It wouldn't make sense to prevent you from attempting to cast the spell simply because it's possible that you will fail to cast it, if it is also possible that you will be able to cast it.