Does dell latidude 3540 (newer version) support an egpu through the m.2 slot?

The only external GPU Dell has ever supported was the Alienware graphics amplifier, which went end of life some time ago and only worked with specific Alienware notebooks.  Dell has never supported eGPUs on any other models.

Bear in mind the shortcomings of retrofitting an eGPU into a notebook without space for the cabling - - you will not be able to reassemble the notebook with the device inside, so you'll be turning your notebook into a non-portable desktop system even if it does work.

If you need a better GPU, instead of buying the parts to convert your notebook into a desktop, look into a good used desktop system.  You'll end up with a desktop anyway, and without the bottleneck of running any GPU through an interface (M.2) that has only a fraction of the bandwidth needed for the GPU to perform up to its capabilities.  And even with a notebook eGPU, you will still need an external screen, as you cannot back feed the GPU signal into the notebook's internal display panel.

The only eGPUs that make any practical sense are those that connect to the notebook by a Thunderbolt external connection.  Anything needing an M.2 internal connector will turn your notebook into a hacked together desktop PC.