Diagnosis
That’s almost certainly grease from a worn-out transmission shaft seal leaking into the wash tub. As the seal degrades, lubricant from the transmission can seep up the agitator shaft and get onto the clothes.
How to Fix It
Unfortunately, this isn’t a simple clean-out fix. It requires replacing the transmission seal and likely the bearings as well, which is a major repair involving disassembling the drive system. For most people, this means a service call.
If That Didn't Work
Before you call a tech, rule out every other possibility. Run a cleaning cycle with a washing machine cleaner and hot water. Wipe down the inside of the drum, the agitator, and the underside of the lid thoroughly with a rag and diluted bleach. If the streaks persist and have a distinct petroleum smell, it confirms the seal diagnosis.
Heads Up
Continuing to run the machine with a leaking seal can lead to more extensive transmission damage and a much more expensive repair. If you see the streaks, stop using it for laundry until it’s fixed.