Most DMMs with 10 Mohms input impedance can be used for this in principle.
Just use the voltage range, e.g. 200 mV instead of the current ranges. This becomes a 20 nA range and you can measure down to 10 pA with a 20000 count meter.
Some (especially true bench) meters have an unexpected high input current up to 1 nA. You can detect this by measuring the open circuit voltage with 10 Mohm input impedance; thus the meters own input current. If this is anywhere above 50 pA (= 0.5 mV), then the meter isn't usable for this purpose.
Of course, the "correct" way to do this is using an electrometer; this allows you to measure down to the femtoamperes...
There's another good way to measure the DMM Input Bias current if the DMM has the usual High Z Input (10G) feature on lower voltage ranges. Us a small 1nF or 10nF quality low leakage film cap (Polypropylene, Polystyrene), place the cap across the DMM inputs and short the cap. Remove the short and time the voltage ramp to a voltage within the range of the DMM setting.
The input bias is calculated as Ibias = C dV/dT, where C is the used Cap, dV is the delta voltage reading and dT is the delta time to reach said voltage. We use this simple method to periodically check our various DMMs to "see" if the input has been damaged (dramatic increase in bias current). Just did so recently with three KS34465A, DMM6500 , AG34401A, HP34401A and SDM3065X, results were (11, 5, 4), 8, 25, 5 & 71pa respectively using a 10nF Polystyrene Cap.
Best,