In the same way that Owen Flanagan wants to naturalize Buddhism by stripping its hocus-pocus, William James focused his attention on personal religious experience rather than the "smells and bells" of traditional institutions. As biographer Robert Richardson puts it, "much of what one usually thinks of as religion James rejects at the start". James says he has no interest in the, "ordinary religious believer, who follows the conventional observances of his country, whether it be Buddhist, Christian, or Mohammedan. His religion has been made for him by others, communicated to him by tradition, determined to fixed forms by imitation and retain by habit." James says he wants to confine himself to "personal religions pure and simple" and say as little as possible about systematic theology or institutional history. The latter are second-hand religions, but he wants to look at the original article. As one might imagine, Richardson says, "James continues to be attacked by church leaders and systematic theologians for his failure to start where they start." James's biographer tells us that this approach to the psychology of religion was a "radical departure, more radical even than that of Friedrich Schleiermacher."
Because James opposed what he called "medical materialism", I suspect that he would be a bit skeptical about neurology as a starting point. This is the view that the value and validity of religious experience can be undermined by classifying the experience as a product of hysteria, fever or by pointing to the mystic's disordered colon. "In the natural sciences and industrial arts it never occurs to anyone to try to refute opinions by showing up their author's neurotic constitution," James says. "It should be no otherwise with religious opinions." I think the idea here is that medical materialism does not seek to explain religious experience so much as it seeks to explain it away. This does not mean he is opposed to scientific inquiry, of course, but to reductionism. On the other hand, I'm sure he'd be thrilled and amazed by the kind of brain imaging technology we presently enjoy. At least one scholar thinks that James and Flanagan would get along quite well.
In her abstract to Bridging Science and Religion: "The More" and "The Less" in William James and Owen Flanagan, author Ann Taves says,"there is a kinship between Owen Flanagan's The Really Hard Problemand William James's The Varieties of Religious Experiencethat not only can help us to understand Flanagan's book but also can help scholars, particularly scholars of religion, to be attentive to an important development in the realm of the 'spiritual but not religious.' Specifically, Flanagan's book continues a tradition in philosophy, exemplified by James, that addresses questions of religious or spiritual meaning in terms accessible to a broad audience outside the context of organized religions." Apparently, there is some overlap and agreement between these two at a general level and it seems that they share more specific affinities with Buddhism.
Unlike Flanagan, the term "Buddhism" doesn't appear in any of James's book titles, but more than a century ago Kitaro Nishida, "Japan's foremost modern philosopher", recognized some very Buddhist ideas in James's radical empiricism, especially James's notion of "pure experience". Joel Krueger's paper, The Varieties of Pure Experience: William James and Kitaro Nishida on Consciousness and Embodiment," explains some of those connections.
David Scott wrote a nice piece that distinguishes the Buddhism that James saw in his own work from the Buddhism that others find in it. In "William James and Buddhism: American Pragmatism and the Orient", Scott goes so far as to claim that the Buddha himself was, like James, a pragmatist and a radical empiricist. His case includes some discussion of John Dewey as well. As "dmf" mentioned recently, there is an interesting paper on Dewey's Zen. The author makes a case that Dewey's emphasis on primordial experience is not some mushy-minded relativism but rather essential "if our thought is to be grounded and transformative".
There is no shortage of this kind of material, but I want to highlight one more example. In "The Strange Attraction of Sciousness: William James on Consciousness" Andrew Bailey sketches out William James’ account of consciousness. Bailey says this account, "has been quite influential in the back-rooms of the recent philosophical and scientific study of consciousness. Daniel Dennett has cited James approvingly," he says, "while Owen Flanagan parades James’ robust notion of a phenomenological stream of consciousness" and "several of the papers in the proceedings of the first major interdisciplinary conference on consciousness ('Toward a Science of Consciousness,' at Tucson in 1994) take James’ doctrines as a central starting point. As work in the burgeoning field of ‘consciousness studies’ reaches fever pitch, James’ thoughts in this area have increased in importance and influence correspondingly."