According to Alexander Lubotsky, there is a pre-Indo-European substratum in proto-Indo-Iranian which can be more plausibly identified with the original language (or languages) of the BMAC, which was, then, eventually given up by the locals in favour of proto-Indo-Iranian.
Moreover, he points out a number of words apparently borrowed from the same language, which, however, are only attested in Indic. Provided this is not an accident of attestation, it may mean that the area where the language (or language family) in question was spoken included at least Gandhara as well, if not the Indus Valley also. This would fit the archaeological evidence mentioned above, pointing to a connection of the BMAC to these areas. Considering that the BMAC is suspected to extend into Afghanistan and Baluchistan (see above), these areas may be included as well.
The assumed Indo-Iranian substratum, then, is potentially relevant to the question about the language of the Indus Valley Civilization, as well.
Alexander Lubotsky, a specialist on Sanskrit at Leiden, has  identified a layer of apparent borrowings in proto-Indo-Iranian which cannot be connected to any known language. The borrowings are united by certain phonotactical characteristics that are untypical for proto-Indo-Iranian; their morphology is opaque from an Indo-Iranian point of view; they have no plausible etymology within Indo-European; and they cover semantic fields specifically connected with the Central Asian environment (such as "camel") and civilization (such as "irrigation canal"). For these reasons, Lubotsky argues that the (problematic, from an Indo-European point of view) etyma he has collected are foreign words borrowed from a substratum, i. e. a language (or family of languages) that has since disappeared, its speakers having given it up in favour of Indo-Iranian.
In view of the semantic characteristics of the loanwords associated with the substratum and the fact that the BMAC is situated square in the center (or perhaps rather at the southern margin, initially) of the probable center of expansion of the Indo-Iranian family of languages, Lubotsky suggests that the substratum language in question is in fact the original language of the BMAC, prior to its assimilation to the Indo-Iranian-speaking Andronovo horizon. This would appear preferrable to an identification of the BMAC as Indo-Iranian-speaking from its beginning given its relatively early attestion (from 2200 BC), its southern location (especially in the case that it indeed covers Afghanistan and Baluchistan as well - see above), and the lack of horse remains.
In addition, Lubotsky points out a sizable number of words with unknown etymology - hence, putative loanwords - known only from Sanskrit - or generally, Indic -, which display the same general characteristics. Provided that the absence of attested Iranian counterparts is not simply due to an accident of history, this seems suggestive in view of the proposed connections of the BMAC to the Indus Valley Civilization.
A possible historical interpretation of this state of affairs runs as follows: From their earliest homeland in the southern Urals and northern Kazakhstan and adjacent areas to the south, where the Indo-Iranians lived in conjunction with Finno-Ugric tribes (there is a considerable number of early loanwords from Indo-Iranian in Finno-Ugric), the Indo-Iranians spread out to the Central Asian steppes, gradually putting a greater emphasis on animal husbandry and extensive pastoralism than even before, especially cattle, as the steppes were unsuited for agriculture. Thriving in the steppes, the population rose and the expansion necessarily continued - eastwards and in particular, southwards - absorbing the thin pre-existing hunter-gatherer population, until, at the southern end of the plains, they encountered a sedentary farming culture, using the Oxus to water their fields through irrigation canals, known to us as the BMAC. The Indo-Iranian cattle herders gradually filtered into the Oxus civilization, eventually assimilating it, and elements from this culture found their way into the newcomers' culture (such as the Soma cult), the two effectively merging. Several groups, perhaps attracted by tales of even greater riches beyond the mountains, spread southeastwards, along the Oxus and into the Hindu Kush, now an area of high linguistic diversity within Indo-Iranian.
Leaving the often barren and cold mountains of eastern Afghanistan behind, where isolated tribes such as in present Nuristan still give hints to this voyage, several treks of cattle herders passed through the Khyber Pass and descended from the mountains into the upper Indus Valley. All along this way, they met natives, some of whom they took as slaves and had them accompany them (as they had probably already done in the Oxus valley), finally reaching the large cities, now partly abandoned, of the Indus Valley Civilization.
(End of Excursus)
The loanwords in Sanskrit, then, seem to point towards the language of the Oxus civilization as including Gandhara as well, and this even raises the possibility that the Indus Valley Civilization was not (or at the very least, not primarily or even exclusively) Dravidian-speaking (pace Parpola and others) and that the (primary) contact with Dravidian (and Munda) speakers occurred not necessarily earlier than the expansion of the Indo-Aryans into the Gangetic plain.
This would be compatible with the view of Josef Elfenbein, a specialist on Balochi and Brahui, who has argued   against the popular view of Brahui, a Dravidian language, as a remnant of the language of the Indus Valley Civilization, and instead advocates a recent migration from the south - likeweise for the other Northern Dravidian languages - based also on the groups' own traditions; it has to be admitted that this does not preclude the possibility that the language of the original inhabitants of the Indus Valley was indeed Dravidian, but has since vanished completely; but at least there is now another candidate for the much-discussed language of the Indus Valley Civilization, and we have reason to believe that the ethnolinguistic landscape of pre-Indo-European South Asia was even more complex.
(See: Dravidian languages, Austroasiatic languages, Veddah language, Burushaski language, the Indo-Iranian substratum language discussed, Kalto language, Kusunda, not to mention the Andamanese languages and the mysterious "Negritos" of Makran).
(Several Tibeto-Burman languages are partly spoken in South Asia, as well, and seem to have a long tradition there).
Whether the Indo-Iranian substratum language and all its (reasonably close) relatives have indeed entirely disappeared, or whether there is a known language continuing it is uncertain; however, from geography considered, the most suggestive candidate would be Burushaski (see also an essay by Michael Witzel, page 6, note 11).
An alternative, half-facetious name for the Indo-Iranian substratum language is "Banāna language", an allusion to the common syllabic pattern short-long-short among words ascribed to it. Words from this language seem to have borrowed into the Sumerian language, as well (Prof. Wolfgang Schulze, instruction).