The Home Secretary appeared on the ITV show on Thursday and said there are "over half a million" doses of the Oxford/AstraZeneca vaccine going out today.
However, Kate wasn't convinced as she pressed Patel if that was true and if more would be going out tomorrow and later in the week.
Priti Patel said: "My understanding is that, with the AstraZeneca vaccine, there are over half a million doses of the vaccine that are going out right now, so that's to vaccination hubs, GP centres, but also hospitals..."
Kate interjected: "So half a million going out today. Or is that half a million over the next week? How quickly will that move from half a million to the two, to the ten, to what we need."
Patel replied: "So that's now. That's absolutely now, then, of course, going up and up and up, because AstraZeneca are accelerating the way they're producing that vaccine and getting it out."
Kate interrupted as she probed: "Sorry that's still not clear. So we've got half a million going out today, what's going out tomorrow? Or is that half a million all we've got going out now?"
Priti answered: "I can't tell you how many we've got going out tomorrow because obviously, we're getting vaccines that are being certified constantly by the authority that does the certification and then does the distribution, so I can't tell you about tomorrow and the day after.
"But we have half a million going out right now and of course those numbers are going up and up and up because the vaccines are being produced..."
Kate returned to GMB on Thursday after having some time off over Christmas.
She revealed she was able to visit Derek in hospital over the festive period but now cannot see him due to the new nationwide lockdown.
Good Morning Britain airs weekdays at 6am on ITV.