You need to get a few things figured out beforehand.
Is the current unipolar, or reversing? That will constrain
device selection and topology. A ground-referred resistor
is the easy case. A truly independent one, is a challenge.
I've made cheesy active loads as you describe but they
were really 3-terminal.
How linear? Across what voltage range? Linearly resistive
through what frequency? These will tell whether you need
an active feedback or a simple control scheme, whether
you have a prayer of operating a FET in linear region to
suit.
What about (say) back-to-back PhotoMOS with the gate
photocells at the center of the butterfly, and a replica
pair in the feedback network of a control loop? That'd be
kinda neat.
But if you have to use "transistors" alone, you'd better have
a nice constrained application-range and attributes.