Reasons why T fares wouldn't be upgradeable into Club:

1. If it's an Avios upgrade, there needs to be availability for an outright Avios booking;
2. If it's a travel agent fare, there cannot be any non-BA metal in the booking (this includes BA Holidays);
3. If it's a travel agent fare, it could be a 'bulk tour' rate which is not upgradeable;
4. As mentioned above, the agent may be trying to upgrade the whole journey, not just the T-class leg.

If there is definitely Avios availability, as opposed to fare difference + change fee availability (which would make no sense for you as it'd be cheaper to cancel) then hang up and call again.

T fares are upgradeable using avios, or at least BA hasn't published anything to the contrary - although there do seem to be a few reports on here of passengers being told that they aren't (although in each case one of the other reasons above applied).