I would suggest that it is within the power of the Thaumaturgy cantrip to create at least a whisper of real words, a real message. Two of the lines of the spell description below support why I think this:
You manifest a minor wonder, a sign of supernatural power, within range.
You create an instantaneous sound that originates from a point of your choice within range, such as a rumble of thunder, the cry of a raven, or ominous whispers.
The first line supports the general idea of being able to do this. Being able to whisper guidance or warning in someone's ear, for instance, would be a minor wonder, a sign of supernatural power. It fits the description, it fits the remit.
As to what you can do within the limits of the spell's power, the second line states that you can produce things such as ominous whispers. Non-ominous whispers are clearly within the scope of "such as". The word "ominous" is used for colour and to refer back to the "sign of supernatural power". I would go further and suggest that if the volume of the sound is up to a rumble of thunder, then normal spoken words would be possible too.
Therefore it is in my opinion both within the remit of the spell and the power of the spell to produce meaningful words close to someone's head within range, perhaps coloured by the domain of cleric casting it.
Of course the range is only 30', anyone else can hear it if they are near enough, if there is background noise it will be as difficult to hear as any other communication and it will be limited in length. I would suggest 3 or 4 words would be on the same scale as a raven's cry, a rumble of thunder or ominous whispers, though that is merely my estimation as a DM.
It has been compared to Message, another cantrip with the express purpose of doing exactly this, suggesting that using Thaumaturgy is undermining it's purpose. However by contrast the range of Message is 120', the target and only the target can hear the message as a whisper, they can reply and you don't need to be able to see the target nor have line of sight, just be familiar with them. Much, much more effective. Also just because one spell impinges on the purpose of another does not make it invalid. Minor Illusion could also be used for this purpose, for instance, and I would not use that as an reason to limit it's usage. There are other examples.
In the end there is nothing, RAW, making it clear if it can or can't be used for this purpose but for the reasons given above as a DM I adjudge it to be a reasonable use for the cantrip in terms of flavour and power in my campaigns, as long as the cleric applies its version of "ominous" :)