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A painless lump under the eye is very likely to be a chalazion. A chalazion is a inflammatory swelling of the modified sebaceous glands present in the eyelids called meibomian glands. These lumps respond very well to warm compresses, given for 10 minutes duration, thrice a day for about 7-10 days. The compresses can be continued for a longer period until it disappears completely. However for persistent swellings which are of the size of a pea , a minor surgical procedure called incision and currettage can be done in the ophthalmologist's office. For very tiny swelling, one can wait and watch. For cosmetic reasons an intralesional steroid injection can be given.
I would suggest that you start with the warm compresses at the earliest and observe for improvement over the next few weeks.
Thank you and best wishes