Typically no. But you will see this become available from more service providers in the coming years. The SMS (texting) network has been integrated with land-line numbers in recent years, but except for specialized uses, that has not translated into broad availability of land-line texting. I would suspect that you will start to see more and more Voice-over-IP (VoIP) providers offering this service in the near future, but since a typical land-line phone isn't really equipped for texting, it will be done through an app instead.
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The phone rings and an auto message says "landline text, press 1 to hear, from (persons #)"
I don't belive that that is possble. try doing it the same way you text to a cell.