Low RPM on processor fan

When/why did you update BIOS?

Did you install the latest BIOS, MS-A08 which was released two years ago (Jun-2020)? 

The MS-A08 support page says: "Fix system will have some fan noise with some particular situations."

Terrible English, but that seems to suggest they reduced fan speeds in MS-A08 to reduce noise...

Yes I had updated the BIOS version to that last one, but I have tried to go back to previous versions and even though I did it did not work, I am currently on the first A00 (2011).

I have tried A-02, A-04, A-08 and still the same.

I even tried with the stock intel fan and it gives similar, this one stays at 1000RPM.

What do you mean "stock Intel fan"? 

There are two fans in the Inspiron 620, the CPU fan attached to the heat sink and a chassis fan.

Try replace cmos battery with a new battery and clear cmos settings via motherboard RTCRST  jumper.

I already changed the BIOS battery, I also tried with the CMOSCLR1 and PSWDCLR1 pins, it tells me that the BIOS was changed but it still shows the same version.

I don't see that jumper you mentioned in the motherboard manual.

I mean that I used the default fan that comes with these processors to check if it was some damage that had the other but still did not reach full speed. The fan I have is the one on the motherboard itself, this one in the image.

The temperature is fine, between 50-67 degrees, which I don't know if it will be fine for an i3-2120, but when it is at 100% usage it heats up the processor and the motherboard, reaching almost like 80°.

If cpu temp is okay I would not be worried about cpu fan rpm.

The CMOS jumper only resets BIOS to its default settings. It doesn't change the version of BIOS that's installed.

If you uninstalled and reinstalled the heat sink with the attached fan, did you gently clean the old thermal paste off top of CPU and bottom of heat sink and then apply a fresh, thin coat of decent thermal paste?

Are all fan blades clean?

Have you used canned air to blow all the dust bunnies off the motherboard?

Are case vents clean and unobstructed (eg, is PC inside a closed cabinet and/or too close to a desk or wall?

Re: The temperature is fine, between 50-67 degrees

that is too high temp for cpu at idle.  have you done a preboot diagnostic PSA test to see if the cpu fan has issue?  Press F12 after power on then choose diagnostic.

I have already done it and it tells me that everything is fine.

Everything is in order, I recently cleaned both the CPU fan and the cabinet fan with a brush, they are free of dust (both are not spinning at high revolutions as before).

I am changing the thermal paste on the processor, but the motherboard is also getting hot.

For a 10-years old system anything can go wrong, those are not designed to last forever and over the time  one way or another electronic components do degrade, that's why you usually won't see warranty for 10 years (if you do - it's normally just replacement service for newer models), PWM subsystem with CPU voltage regulator or even BIOS update circuit components are not an exception, any sneeze can trigger chain of failures. In a world of IT - archeology or necromancery are not a general trend. 

I had forgotten to keep reporting, so after all this time what I did was to give a cleaning to the whole computer and change the thermal paste, now appears system fan failer, but well there is not much to do because perfectly both cpu and cabinet fans rotate well.

Then with the RPM and was solved and returned to be as before, now with this at maximum use the CPU does not go from 59 ° at maximum capacity, and about 39-48 ° goes between use at 0% and 50%.

I have a dell laptop with inspiron13 and the processor fan have 0 RPM