What you have are bearing that have gone bad.... The greasy substance is just leaking gunk build up of dirt, gelled fabric softener & liquid detergent and a little bit of lub grease from the tub seal.
Your tub seal has failed and letting the above ingredients into the bearings allowing them to rust and fail.

Here is the part you will need.
Whirlpool W10435302 Bearing and Tub Seal Kit - AppliancePartsPros.com
It is a long sometimes difficult job and requires a special bearing press tool.
The above link includes a how to video.
If you choose to do the repair you can find places online to rent the tool.

If you are up to it I recommend doing it as this is a great machine, I rebuilt a trade-in for my wife, who uses it with 2 to 3 loads a day.

As always I recommend to anyone with a 110/120 volt washer that has a computerized / electronic control module, to protect it with a power surge protector, like you would use for you PC!