Middle English reke "smoke, fumes; steam, vapor," from Old English rec (Anglian), riec (West Saxon), "smoke from burning material," probably from a Scandinavian source such as Old Norse reykr, Danish rǿg, Swedish rök "smoke, steam."
These are reconstructed to be from Proto-Germanic *raukiz (source also of Old Frisian rek, Middle Dutch rooc, Old High German rouh, German Rauch, Icelandic reykr "smoke, steam"), from PIE *reug- "to vomit, belch;" also "smoke, cloud."
The sense of "stench" is attested 1650s via the notion of "that which rises" (compare reek (v.)). Century Dictionary (1891) marks the word "Obsolete, archaic, or Scotch." According to OED, "As the word has chiefly survived in northern use the palatalized form reech is comparatively rare." A c. 1250 document refers to the period March-April as Reke-fille "the misty month."
Middle English reken "to emit smoke," of smoke or stench, "to rise," from Old English recan (Anglian), reocan (West Saxon) "emit smoke," from Proto-Germanic *reukan (source also of Old Frisian reka "smoke," Middle Dutch roken, Dutch rieken "to smoke," Old High German riohhan "to smoke, steam," German rauchen "to smoke," riechen "to smell"); from the same source as the nouns (see reek (n.)).
Originally a strong verb, with past tense reac, past participle gereocen, but occasionally showing weak conjugation in Old English. Meaning "to emit a bad smell" is recorded from 1710 via sense "be heated and perspiring" (early 15c.). Related: Reeked; reeking.