We present conjectures for the space-time form and leading large-N dependence of extremal and near-extremal correlation functions in the N = 8 superconformal Yang-Mills theory in d = 3 as well as in the (0,2) superconformal theory in d = 6, using their gravity duals with M-theory on AdS₄ × S⁷ and AdS₇ × S⁴ respectively. As a key part of the conjectures, we argue that the bulk couplings associated with extremal and near-extremal field configurations in the corresponding AdS₄ and AdS₇ gauged supergravities vanish. The vanishing of these couplings constitutes a generalization of the property of consistent truncation of the Kaluza-Klein modes.