The 46-year-old model has helped to popularise the expression - which refers to smiling with your eyes - over recent years, but her efforts to have the word added to the Merriam-Webster dictionary have been consistently rebuffed by the publishing firm.
Speaking about her big ambition, Tyra explained: "There's a secret smize. There's a lot of smizes. But you know what? Guess who will not put smize in the dictionary? Merriam-Webster.
We keep calling the Merriam-Webster people."
Tyra revealed that she and her team have inundated the company - which publishes the American dictionary - with calls and emails, but they have so far been unsuccessful.
The model told NPR: "We call them. We email them. We show them the cover of The Wall Street Journal.
"We keep - like, we show everything, all this stuff. And they're just like, oh, you know, we've been - we've had our eye on smize for a couple of years. And I'm like, you know what? Now you're just hating."
Tyra also explained that smizing can express a broad range of emotions.
She said: "There's, like, different ones. And you have to think of different things to create a different - to activate a different muscle in your face."
The new Dancing with the Stars host has created her own ice cream company called SMIZE Cream during the coronavirus pandemic.
Tyra initially planned to launch her new business venture in 2021, but she chose to accelerate her ambitions because she felt "people need ice cream" amid the current crisis.