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"rabbit-hole, fox-hole, etc.," c.1300, borewe, from Old English burgh "stronghold, fortress" (see borough); influenced by bergh "hill," and berwen "to defend, take refuge."
c.1600, "to place in a burrow, from burrow (n.). Figuratively (e.g. to burrow (one's) head) by 1862. Intransitive sense, "to bore one's way into, penetrate" is from 1610s, originally figurative (literal sense, of animals, attested by 1771). Related: Burrowed; borrowing.