There is no specific rule for this, or it would say so in the spell description. All it says it that the creature hears the message in its mind, which for the purposes of being asleep wouldn’t be that different from somebody speaking to you while sleeping.
Though there is an argument that the sleeping creature is more likely to remember and consciously register the words since the creature definitely “hears the message”.
Whether this is enough to wake them… I think a DM would just have to either decide whether the creature is a light enough sleeper or possibly allow some kind of check (a Wisdom check?).
I would also say that they would have to be awake and conscious to reply “in a like manner” since you cannot take actions while asleep.*
*Unless replying “five more minutes, Grandma Frog” would be funnier, of course.