This issue has been hotly debated by believers throughout history. Many, like the Lutheran commentator R. C. H. Lenski, see no difficulty with Satan’s miraculously transporting Jesus to the heights of the temple:
Like Job, Jesus was placed into Satan’s power so that the latter might tempt him to the uttermost. The transfer of Jesus to the Temple was a physical transfer. There is no difficulty as to willingness on Jesus’ part; he consented to the Father’s will to be tempted as the devil might will to tempt him. We need not say that Jesus transferred himself to the Temple; paralambanei and esteise indicate that the devil provided the motive power. Throughout, Jesus only submits to the tempter’s operations. The devil was permitted to take Jesus where he desired for the purpose of temptation.
On the other hand, one aspect of this account implies that Satan may have appeared to Christ in a vision. In verses 8 and 9, we read that from a high mountain he showed Jesus all the kingdoms of the world in their splendor. William Hendriksen points out that even if we grant that Satan was granted the power by God to physically transfer Jesus to these dangerous and inaccessible locations, “some kind of miracle (would) have been required . . . to show Jesus all the kingdoms of the world, and this not just in dim outline, but very distinctly, so that ‘all their splendor (or glory)’ would be plainly visible; and again, not little by little during a lengthy period of time, but, as Luke adds, ‘in a moment’ ” (The Gospel of Matthew, p. 231).
Because of the clearly visionary nature of verses 8 and 9, it is likely that Satan was permitted by God to supernaturally tempt Christ in a vision. However, because some good arguments can be marshaled in favor of Christ having been physically transferred by Satan, it probably wouldn’t be wise to be dogmatically committed to either interpretation.