The TL081 is one part that will work. It would likely need to be nulled.
The TL081 and similar JFET operational amplifiers will work. A TL031 or TL051 would have a lower offset voltage and is a better choice.
In the old days, they used the LM301A bipolar part and it is still a good choice. Of course now there are lots of parts available with an input common mode range which includes the positive supply although they are a little more expensive.
Nevertheless I was not expecting 0,5V this is much larger than what the datasheet states (20mV), or am I interpreting it wrong?
The datasheet graph for negative output saturation voltage versus current shows the problem. The LM324/LM358 can only actively pull down to about 0.5 volts with a PNP emitter follower. The rest of the current is provided by a 50 microamp current sink down to 20 millivolts or so.
As shown below in a full schematic of the LM324, Q11 pulls down to 0.5 volts and Q10 pulls down with 50 microamps below that.
The LM1014 is an improved drop in replacement for the LM324 which can sink current all the way to ground. The LT1013 is an improved replacement for the LM358.
The circuit Xavier60 suggests is a better way to do this.