Can you tell me about the leather? If I use terms you dont understand just google them and check the spec sheet of your bag online.
Is it a pigmented or aniline finished bag? Does it have a plastic feel to the surface? Sometimes leather can bloom where fats used in the tanning can come to the surface. This would be unusual if pigmented with a surface finish.
Use a hairdryer and carefully warm the leather (if it's too hot for your hand, the bag won't like it either!) Around the marks, don't use the hair dryer for too long in one spot. If it is fat bloom then it should soak back into the leather. Try this first then we can eliminate that possibility and move on to the next.
Usually I would use Saphir's Renomat cleaner to remove stains on my leather goods. If that doesn't work because the stain is quite old, I would use their black cream polish to cover the stains - works all the time, especially for black leather.