Most Kenmore washers are made by Whirlpool. From what I understand, it would be unusual for such a washer design to leak transmission oil into the tub where it can come into contact with clothing.
The only way to tell for sure which manufacturer made this particular washer is with the model number off the model and serial number tag. You can find tips for locating the model number tag on your appliances in the 'Repair Parts' section of my site linked below.
The Appliance Information Site