Middle English swimmen, from Old English swimman, of a person, fish, bird, "to move in the water, float on the water, move in water by natural means of locomotion" (class III strong verb; past tense swamm, past participle swummen), from Proto-Germanic *swimjan (source also of Old Saxon and Old High German swimman, Old Norse svimma, Dutch zwemmen, German schwimmen), from PIE root *swem- "to be in motion."

The root sometimes is said to be restricted to Germanic, but according to OED possible cognates are Welsh chwyf "motion," Old Irish do-sennaim "I hunt," Lithuanian sundyti "to chase." The more common Indo-European root is *sna-.

The transitive sense of "cross by swimming" is from 1590s. Figurative phrase sink or swim is attested from early 15c. The figurative use of swim with (or against) the tide is by 1590s.

The sense of "reel or move unsteadily" is recorded by 1670s in reference to objects seen by a dizzy person; in reference to the head or brain, "be affected by dizziness, have a giddy sensation," from 1702. Compare archaic noun swim "a dizziness, swoon, trance," from Middle English swime "unconscious state," from Old English swima "unconsciousness." The notion appears to be "a swimming in the head." Chaucer has swimbel "giddy motion."