word-forming element meaning "not, opposite of, without" (also im-, il-, ir- by assimilation of -n- with following consonant, a tendency which began in later Latin), from Latin in- "not," cognate with Greek an-, Old English un-, all from PIE root *ne- "not."
In Old French and Middle English often en-, but most of these forms have not survived in Modern English, and the few that do (enemy, for instance) no longer are felt as negative. The rule of thumb in English has been to use in- with obviously Latin elements, un- with native or nativized ones.
early 15c., "fit or proper to be chosen," from Old French eligible "fit to be chosen" (14c.), from Late Latin eligibilis "that may be chosen," from Latin eligere "choose" (see election). Related: Eligibly.
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Harper, D. (n.d.). Etymology of ineligible. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/ineligible