When the DSO138 is set to the highest sensitivity range, i.e. SEN1 = 10mV there is DC offset of about 10mV on the displayed trace which cannot be corrected.

Here is the schematic of the input preamp stage.



The input uses a TL084 quad JFET opamp that has input resistance of 10^12Ω.
Typical input offset current = 5pA
Typical input bias current = 20pA

When SEN1 = 1V
R5 = 20kΩ is the effective shunt to GND.

When SEN1 = 0.1V
R3 = 200kΩ is the effective shunt to GND.

When SEN1 = 10mV
the effective shunt to GND is the combination of R2 + R3 = 2MΩ
in parallel with R4 + R5 = 2.02MΩ
The net effective shunt to GND is about 1MΩ

One possible fix would be to inject a small offset current into pin-1 at the SEN1 switch.



On the top side of the board there is easy access to AV-1 and AV+ .
Solder the outer leads of a small trimmer pot to these two solder points, leaving the center terminal free. Any pot higher than 10kΩ would be suitable.
Solder a 10MΩ (or even higher) resistor between the center lead of the pot and where you see R1 and C2 meet on the PCB trace leading to pin-1 of SEN1 slide-switch.



This mod takes effect only when SEN1 is set to 10mV.

To adjust, disconnect any cable input to the scope.
Set:
CPL = DC or AC
SEN1 = 10mV
SEN2 = X1

Adjust the trimmer pot to bring the trace to the center of the screen.

 

I wonder if it would help to put a resistor between pins 6 and 7 of the TL084? 1 Meg maybe?

 

Is the board clean around the input stage? If not give it a clean.
Maybe the TL084 input has been damaged and is more leaky than it should be - change the TL084.

 

I wonder if it would help to put a resistor between pins 6 and 7 of the TL084? 1 Meg maybe?

Trying to "cancel" input bias current effects with FET-input op-amps is generally an exercise in futility and simply adds noise.

The TL084 is a rather old design and no great shakes in terms of DC performance. 10 mV of offset is above typical but below maximum for the standard version at 25 °C. The B suffix part has the best offset spec in the commercial temperature range part. I'd be inclined to start with replacing the existing amp with a B version (assuming it isn't already). There is probably some newer amp with lower offset that could replace it, but I can suggest one offhand. Other than DC performance, that series is not too shabby.

With the input grounded the impedance at the amp inputs is really quite low, so I would not suspect board contamination as a big contributor. If the offset changes considerably if you terminate the input with say 10 megs, then I would suspect contamination.

You could inject an offset voltage as you propose. It will of course cause some attenuation of the input signal and it will be affected by the source resistance. Usually offset is injected into the amp on the "opposite input" from the signal, but that makes trying to cobble it in a bigger undertaking, especially with a non-inverting amp with unity gain.