The vacant set of random interlacements on ℤd, d≥3, has nontrivial percolative properties. It is known from Sznitman [Ann. Math. 171 (2010) 2039–2087], Sidoravicius and Sznitman [Comm. Pure Appl. Math. 62 (2009) 831–858] that there is a nondegenerate critical value u∗ such that the vacant set at level u percolates when u∗ and does not percolate when u>u∗. We derive here an asymptotic upper bound on u∗, as d goes to infinity, which complements the lower bound from Sznitman [Probab. Theory Related Fields, to appear]. Our main result shows that u∗ is equivalent to log d for large d and thus has the same principal asymptotic behavior as the critical parameter attached to random interlacements on 2d-regular trees, which has been explicitly computed in Teixeira [Electron. J. Probab. 14 (2009) 1604–1627].