If you're talking about a place, like where something is located, use 'premises'. 'On-prem' and 'Off-prem' are shorthand for "on/off the premises".

If you're talking about a declarative statement used as the foundation of an argument, use "premise".

Examples:

"Were those databases moved to the cloud?"

"No, they're still on the premises." (Premises = place)

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"Why did we decide to move those DBs to the cloud?"

"On the premise that it's cheaper than managing the server administration ourselves." (Premise = something stated as the basis for an argument)

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"Can you give me a few more reasons that we moved those DBs to the cloud for the executive committee meeting?"

"We made the decision on the premises that it's cheaper to run, our basement server room regularly floods, and our current server admin is a drunk." (Premises = plural of premise)