An Etymological Dictionary of the German Language/dürfen
dürfen, anom. vb., ‘to be allowed, venture, need,’ from MidHG. dürfen, durfen, a preterite pres., ‘to have reason, cause, be permitted, need, require’; OHG. durfan, preterite pres., ‘to lack, be destitute of, require, be in need of’; comp. Goth. þaúrban, Du. durven, AS. þurfan, ‘to be in need of.’ In addition to the Teut. root þurf, þurb, Swiss points to an old parallel form þurp. In the ModHG. deriv. darben, Bedürfnis, Notdurft, bieder, &c., the primary sense of the root þrf, from tṛp, ‘to be destitute of, lack,’ still appears.