LM317T as a current regulator
Datasheet check!

IADJ: Adjustment pin current 


Typical 50 µA
Maximum 100 µA


So... it will not work.

For an accurate 10µA you will need to use a different topology.


Opamp-based current source
FET/JFET/MOSFET as a pass element, controlled by opamp. Bipolar could also be used, depending on required temperature drift.


Choice depends on factors you will need to mention, like how much parasitic capacitance you can tolerate at the output of your current source, voltage compliance, available supply voltage, etc.