Here is my proposed solution:
In this blog post I find instructions for enabling nested virtualization instructions for ESXi or VMWare Workstation v8 on a Windows host. Though the blogger cites installing Microsoft's HyperV in his guest as his purpose for doing so, the same configuration should pass the necessary cpu instructions on to your guest's kvm module. Relevant screenshot:

The same blogger warns us:
Remember nesting a hypervisor means it’s going to run very, very slow...
And I suggest it will (at least) work because...
I know the Kernel's Virtual Machine requires hardware-enabled virtualization CPU extensions. So the only way to enable kvm is to present it with either AMD-v or VT-x cpu extensions depending on your CPU's vendor, of course. Apparently, according to this, since vSphere 5 it is possible to present this in a virtualized 64-bit guest and therefore nest virtualization hosts. Since you only mention "VMWare" it's impossible for me to know exactly which of their products it is you're attempting to use - each has different capabilities.
But here are instructions for installing VMWare's ESXi in VMWare Workstation on a Windows host. Just being able to do so leads me to believe it must also be possible for kvm.
My own AMD architecture has supported AMD-Vi - or nested virtualization to include IOMMU - for years. According to this the kvm features that can make use of it are enabled by default, but the same is not true for Intel architecture. It seems on Intel architecture the kernel will ignore the necessary CPU extensions if you dont hand it this parameter at boot:
kvm-intel.nested=1
Alternatively you could configure the kvm module itself via modprobe by adding this to /etc/modprobe.d/dist.conf:
options kvm-intel nested=y
The above is primarily focused on how to enable a kvm host to pass virtualization instructions on to its guest - in other words doing the above should be necessary on a host machine but not in the guest. The guest should detect the extensions if present and enable the module - if installed - by default. But most noteworthy to me is that it is possible.
Going back to 2 and I find this addendum toward the bottom of the page:
If you are using an i3 or later processor (that is, you do not have a Core 2 Duo), you can enable nested Virtualization Technology (VT). This allows you to run 64-bit virtual machines within ESXi. To enable nested VT, add this line to the .vmx file of the ESXi virtual machine:
vhv.enable = "TRUE"
Then again, maybe all of your software supports propagating VT-x/EPT and/or AMD-Vi and the only missing piece of the puzzle is that either your CPU doesn't (possible) or that you haven't enabled it in your system's firmware (a little more likely).