"would" is missing out of both of those sentences. You might hear this spoken as, "I like to know...", but it should be "i'd like to know..." or "I would like to know..."
when it's spoken, it should be "I'd like..." or "I would like..."
As for which one is correct, it depends on what you mean.
O - I'd like to know if the rumor is true or not (this is correct, and is asking about one rumor, show they want to know if it's true or not true)
O - I'd like to know if the rumors are true or not (same as the example above, but about multiple rumors)
X - I'd like to know if rumors are true or not (this is technically correct, but probably not what you want to say. it's not natural. And means someone wants to know if any rumors about anything are correct. It's too vague to understand the meaning.