During his final days in Geneva, Borges began brooding about the possibility of an afterlife. Although calm and collected about his own death, Borges began probing Kodama as to whether she inclined more towards the Shinto beliefs of her father or the Catholicism of her mother. Kodama "had always regarded Borges as an Agnostic, as she was herself", but given the insistence of his questioning, she offered to call someone more "qualified". Borges responded, "You are asking me if I want a priest." He then instructed her to call two clergymen, a Catholic priest, in memory of his mother, and a Protestant minister, in memory of his English grandmother. He was visited first by Father Pierre Jacquet and by Pastor Edouard de Montmollin.
Borges died of liver cancer on 14 June 1986, aged 86, in Geneva. His burial was preceded by an ecumenical service at the Protestant Cathédrale de Saint Pierre on 18 June. With many Swiss and Argentine dignitaries present, Pastor de Montmollin read the First Chapter of St John's Gospel. He then preached that "Borges was a man who had unceasingly searched for the right word, the term that could sum up the whole, the final meaning of things." He explained, however, that no man can reach that word through his own efforts and in trying becomes lost in a labyrinth. Pastor de Montmollin concluded, "It is not man who discovers the word, it is the Word that comes to him."
Father Jacquet also preached, saying that, when visiting Borges before his death, he had found "a man full of love, who received from the Church the forgiveness of his sins". After the funeral, Borges was laid to rest in Geneva's Cimetière de Plainpalais. His grave, marked by a rough-hewn headstone, is adorned with carvings derived from Anglo-Saxon and Old Norse art and literature.