yes it supports.  Dell does not offer it because 3040 is entry level low tier compared to 7040.  in general dell does not offer high end configuration for low tier model.   Dell OptiPlex 3040 Performance Results - UserBenchmark Popular components in PC builds with the Dell OptiPlex 3040 Motherboard.
I know the official specification does not show the i7-6700 CPU as an option, but will the Optiplex 3040 support this CPU? I've seen someone claim that he bought an Optiplex 3040 with an i7-6700. Can this be done? The Intel chipset does seem to support it, but does the BIOS have the microcode built in to accept this CPU? Has anyone tried it in the past?
Does an Optiplex 3040 support i7-6700 CPU?