Old English felan "to touch or have a sensory experience of; perceive, sense (something)," in late Old English "have a mental perception," from Proto-Germanic *foljanan (source also of Old Saxon gifolian, Old Frisian fela, Dutch voelen, Old High German vuolen, German fühlen "to feel," Old Norse falma "to grope"), which is of uncertain origin, possibly from a PIE *pal- "to touch, feel, shake, strike softly" (source also of Greek psallein "to pluck" the harp), or from PIE root *pel- (5) "to thrust, strike, drive."
In Germanic languages, the specific word for "perceive by sense of touch" has tended to evolve to apply to the emotions. The connecting notion might be "perceive through senses which are not referred to any special organ." Sense of "be conscious of a tactile sensation, sense pain, pleasure, illness, etc.; have an emotional experience or reaction," developed by c. 1200, also "have an opinion or conviction;" that of "to react with sympathy or compassion" is from mid-14c. Meaning "to try by touch" is from early 14c. From late 14c. as "know (something) beforehand, to have foreknowledge of." To feel like "want to" attested from 1829.
late 14c., "that can be felt, perceptible by the touch," from Late Latin palpabilis "that may be touched or felt," from Latin palpare "touch gently, stroke," a word de Vaan finds to be of no known etymology (rejecting the connection in Watkins, etc., to a reduplication of the PIE root *pal-, as in feel (v.), on phonetic grounds). Some sources suggest it is onomatopoeic. The figurative sense of "easily perceived, evident, clear, obvious" also is from late 14c., on the notion of "seeming as if it might be touched." Related: Palpably; palpability.