In The Matrix Reloaded the Oracle confirms to Neo that she is a program. She considers it "obvious" that she is a program, probably because she is clearly too old to be a human. Even Morpheus knows that she is very old:
Neo: So is this the same oracle that made the, uh, prophecy?
Morpheus: Yes. She's very old. She's been with us since the beginning.
Neo: The beginning?
Morpheus: Of the Resistance.
Morpheus also thinks that the Resistance has fought the machines for about a century:
I remember that for 100 years we have fought these machines...
Taken together, these quotes from Morpheus imply that the Oracle is at least 100 years old. Even if Morpheus rounded up when he said they've been fighting for 100 years (maybe they've only been fighting for 80 years), the Oracle would still have to be too old to be human. Even if her shell showed signs of aging, she would have had to appear as a baby at the "beginning of the Resistance" and yet she still looks too young to be as old as she'd have to be.
Moreover, the Oracle never leaves the Matrix (she can't, since she doesn't have a human body) so if Morpheus thinks she is human then from his perspective she's basically a blue pill who is an important member of the Resistance yet for some reason refuses to leave the Matrix.
It indeed seems obvious that the Oracle isn't human. But Morpheus seems to think she is, otherwise why would he trust her? The next time Morpheus sees the Oracle (after she's confirmed to Neo she's a program) he says:
Morpheus: After everything that's happened, how can you expect me to believe you?
The Oracle: I don't expect you to do anything. I expect what I've always expected, for you to make up your own damn mind. Believe me or don't.
It sounds like now he realizes she's a program (although he could be referring to all the other things that happened). So when did Morpheus realize the Oracle is a program? Did he realize it before he took Neo to see her? Or did he not find out until she told Neo that she's a program?