It will definitely work. I am attaching a picture of RTX 3080 installed in DELL Precision 5820

It will definitely work. I am attaching a picture of RTX 3090 installed in DELL Precision 5820.
Remember that your PSU must be at least 850W, preferably more.

It would be best to borrow from a well-known RTX 4090 GPU and try it out!!!

The RTX 4090 = 450w.  Of the PSU options available for the 3660, you'd need the 1000w PSU.  The RTX 4090 requires a 16-pin connection that you might want to stay away from for now due to melting issues.

Have you purchased the 4090 yet?

Hi there, can you elaborate more on the melting issues?

RTX 4090 connector melting issues 

I purchased the 4090.  It will not fit into the Dell Precision 3660.

The NVIDIA GeForce RTX 4090 will NOT fit into a Dell Precision 3660. I bought both after reading this thread and was sadly disappointed to find out that the NVIDIA GeForce RTX 4090 doesn't fit into the Dell Precision 3660. The card presses against the power supply so it cannot fit into the proper port. Also, with the version of the card I purchased, PNY, you couldn't bend the power cables enough to fit the cover on the case.   

Technically the 4090 can be converted with a kit, to single slot

https://www.notebookcheck.net/World-s-smallest-RTX-4090-card-Alphacool-s-latest-waterblock-transforms-the-Nvidia-beast-into-a-compact-1-slot-GPU.687073.0.html

but there would be still the issue of adding a radiator. I don't see it managing to fit one inside. outside maybe....