If you prefer programming, get an ADC that can read +/- 10V, read it with an Arduino, then get a 4-20mA driver IC, something like the AD5749 or similar. Depending on the speed and number of channels probably any low end Arduino should be capable.
If you prefer analog then you will need a bipolar supply (+/- 12 or 15V) and probably two stages of
level shifting circuit. You might be able to do it all one stage, but I'd have to go through the design process to figure out how to do it. It's probably easier to shift the signal to 1-5V and then use a 4-20mA driver circuit.