When I was a child, I must have read the whole series through half-a-dozen times. When a schoolmate pointed out to me the parallels between Aslan and Jesus, though, I felt cheated; it was as though Lewis were smuggling in propaganda.

“The Lion, the Witch and the Wardrobe is very clearly a book about the life of Christ,” says Williams in his quietly resonant voice, “in a way that isn’t true of the others.” I ask why Williams does not write much about the famous scene in which Aslan is killed and miraculously resurrected. “I think it is such an obvious parallel,” he says. “The more interesting thing is how does Lewis convey a sense of what the religious climate, the religious sensibility might be in another world? That is the teasing thing.”

Why turn Christ the lamb into Christ the lion? In his autobiography Surprised by Joy, Lewis writes about the exhilaration of his conversion from atheism to Christianity. There is a feeling, says Williams, “that something really quite fierce has taken hold of people” when they turn to God. “I’ve always connected it in my mind with T S Eliot’s image of Christ the tiger in ‘Gerontion’ – something springs on you.”

Fearsomeness is not a quality many people associate with the Church of England – or indeed Williams. His evangelical enemies accuse him of being too wishy-washy and open-minded – the same qualities that have led some secular intellectuals to claim him. (As one atheist put it to me: “We know he secretly doesn’t really believe in all that Jesus and God stuff. He just thinks they’re excellent metaphors.”)

Both sides fundamentally misread the Archbishop. In a recent review of the biblical scholar Géza Vermes’s Christian Beginnings, which traces how the doctrine of the incarnation developed over time, he writes respectfully of a “beautiful and magisterial book”. Yet he is firmly convinced that Jesus’s followers regarded him as divine long before the Council of Nicaea decreed it in 325; and he writes with awe of “the divine life being clothed in human flesh and blood”. He is certain the tomb was empty.

The place for doubt in Williams’s world is within religion. Of all his works, he admires most the short book Lewis wrote after the death of his beloved wife Joy, A Grief Observed. He calls it a “staggering” book, far angrier than his other Christian writings. When you are happy, Lewis writes, God welcomes you. “But go to Him when your need is desperate, when all other help is vain, and what do you find? A door slammed in your face, and a sound of bolting and double bolting on the inside. After that, silence.”

Remarkably, similar struggles are present in Narnia. In The Silver Chair, a witch has imprisoned the marsh-wiggle Puddleglum with Eustace and Jill. She tries to persuade them the world above ground is only a fantasy. The children are ready to give in but Puddleglum resists: “I’m on Aslan’s side even if there isn’t any Aslan to lead it.” Those lines seem remarkably close to the theories of some modern theologians who hold that the truth of religious language is not as important as how useful it is in everyday life. We’re back to the idea of God being an excellent metaphor.

Williams doesn’t agree. “Puddleglum’s great statement of faith isn’t saying it doesn’t matter whether it’s true or not. He’s saying I have no means of knowing whether this is or isn’t true… But I know there’s something here that I can’t let go of without letting go of myself.” He compares Lewis to St John of the Cross, the Spanish mystic who went through a dark night of the soul but came through to see the light of truth. “Lewis thought most theologians were gutless liberals who didn’t care about the truth enough.” (Do I detect a raised eyebrow on the phrase “gutless liberals”?)

One of the most vivid scenes in the series comes in The Voyage of the Dawn Treader when Eustace – the spoiled child of non-smoking teetotal vegetarians: never a good sign in Lewis – is turned into a dragon. He tries to peel off his skin but finds only another set of scales. It takes Aslan to cut his claws in deep and rip it off – a “feeling worse than anything I’ve ever felt”, as Eustace says – for him to be reborn.

Williams thinks the sequence is astonishing. “We are all of us human beings in love with fictions about ourselves – dramas in which we star. And because Lewis was a big dramatic character himself, I think he knew it as well as anyone.” Self-examination will, however, only take you so far. “Eventually Eustace has to turn to Aslan and say – go on, you do the job, and he finds layers he never knew he had being stripped away.”

Most children, I imagine, miss the allegory; but most also enjoy getting their backs scratched, and that physical sensation is what they respond to. Aslan’s vital sensuousness, which Williams goes as far as to say is “on the knife-edge of the erotic”, is also on display in the passage in The Lion when Lucy and Susan roll around in the grass with the big cat. “Lewis in all his writing has a very strong sense of physical pleasures,” Williams tells me. “All kinds of physical pleasures, the pleasure of a pint in the pub, which he notoriously celebrates on many occasions.”

The Narnia books were written in a very different era from our own and are now attacked for being misogynist – remember that notorious moment when Susan is excluded from Narnia because of loving “nylons and lipsticks and invitations”? – and even racist. Aslan’s swarthy enemies the Calormens speak like parodies of Arabian Nightscharacters, and worship Tash – a bizarre concoction of Babylonian devil and Hindu god. Lewis wrote in his letters that The Horse and His Boyshowed “the calling and conversion of the heathen”. Some have even pointed out that the White Witch tempts Edmund with Turkish Delight.

Williams denies none of this. He points out, though, that Lewis was more flexible in his attitude to other religions than you might expect. In The Last Battle we meet a Calormene soldier called Emeth. All his life he has served the diabolical Tash and when he confronts Aslan you might expect him to be cast into hell. Instead Aslan says “unless thy desire had been for me thou wouldst not have sought so long and so truly” (a near direct quotation from St Augustine) and brings him into heaven. “Here is someone with total courage, passion and generosity,” says Williams, “who’s giving all that to a mistaken target. But the heavenly postman knows better and delivers it to the right address.”

Before I land Williams in trouble (hold the front page: Archbishop says pagans go to heaven), I don’t think he believes other religions are equivalent to Christianity. But if, as he tells me, “grace is everywhere”, it is likely to turn up in unexpected places. Here we are not only talking about religion: what about people who serve oppressive governments or work in morally compromised professions? Can you occupy any position of authority and remain untainted?

These are surely questions the head of the established church must ponder. His interventions in live controversies – saying that aspects of Islamic sharia would inevitably become part of British law, or describing the Big Society as “aspirational waffle” – show he is unafraid of disturbing the mainstream. But I wonder whether his statements on the temporal issues of the day may have drowned out his spiritual message. His role might be in part a political one, but he is more provocative when he tackles the deeper questions.

He leaves his position as Archbishop at the end of this year to become Master of Magdalene College, Cambridge – C S Lewis’s old college. Doubtless he will return to his academic interests, but it would be a shame if he retreated to speaking only to other theologians. In The Lion’s World his sometimes knotty prose style relaxes into an inspiring clarity. The ideas stay with you long after you finish the book, and his parting words on Lewis could apply equally to him. Great writers, he tells me, provoke you into looking beyond yourself. They seem to say: “Do you recognise that? Does that ring a bell? Something is moving in on you – well, getting its claws into you.”