I found this on but Im not sure if it is what I am looking for:

a staring gaze; a fixed look with the eyes wide open: The banker greeted him with a glassy stare.

May I say “That girl has an intense stare”? or “That girl has an intense gaze”?

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Both those are possible. They do not always have exactly the same nuance; “stare” can feel more intrusive.

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