Yes, surprising as it is, I found credible sources indicating that there was some discussion of offering India East Africa as a mandate.
Perhaps it is useful for others who wish to read more about this a full detail of my sources. In "How India Became Territorial: Foreign Policy, Diaspora, Geopolitics (2014)" by Itty Abraham, I found this quote:
On war reparations, in spite of their considerable contribution to the
war effort, the opinion of lesser powers and British dominions would
count for little. Powerful elite Indian opinion had proposed that India
be permitted to take over the German colonies in East Africa as a
mandate, but that did not happen.
Abraham references "India and East Africa: Imperial Partnership at the End of the First World War (1971)" by Herbert Lüthy, adds some important context:
It is in this context that the even more curious question of a League
mandate for India came up, rather casually, for deliberation in the
British War Cabinet in 1918, and again in the Mandates Commission on
African Colonies at the Paris Peace Conference. With the exception of
Canada, each British Dominion was ultimately entrusted with mandatory
powers over some part of the colonial spoils from Germany's lost
empire; to entrust India, or rather the Government of India, with a
mandate over former German East Africa might have added some more
substantial bribe to the purely symbolic promotion to League
membership, although - quite apart from all the incalculable
consequences we can envisage from hindsight - it would equally have
added a further paradox to the ambiguity of imperial and international
status. League membership for India had been difficult enough to
admit, but had been allowed for as making simply 'an additional voice
for the British Empire' (the founding of the United Nations witnessed
a similar allowance for another great power), but a colonial mandate
for India would have been more than even the most subtle casuists of
international law could stomach.
He goes on to quote "The International Status of India (1931)" by Lanka Sundaram, which provides the closest secondary evidence I could find of this discussion:
The League Mandate for the administration of German East Africa (now
Tanganyika) was at one moment on the point of being granted to India,
but this courageous step, which would have enhanced the value of the
juridical basis of India's international status, was at the last
moment withdrawn in favour of Great Britain.
Finally, I did manage to dig up a primary source document indicating the same thing. In the "Minutes of the 37th Imperial War Cabinet, Minute 8" (a 50 MB PDF), the British clearly intended to at least allow India to make its wishes known with respect to German East Africa.
Sir Robert Borden said that there were still a further question on
which it was possible the British Government would differ from the
American Government, and this question was with regard to the
retention of the German colonies. Sir Robert reminded the Imperial War
Cabinet that he had already suggested by telegram that those
dominions who were principally interested in this question should put
forward their views...
... the view of the British Government was that none of the German
colonies should be restored, and that those German colonies which had
been captured by colonial troops, with the possible exception of
British East Africa, should be held by the dominions which had
... the Imperial War Cabinet decided that - At the important Allied
Conference which should precede the Peace Conference, India and each
self-governing dominion should be given the fullest opportunity to
express their views on those questions which may closely concern them.
I think that the evidence strongly indicates that such an offer was at least (perhaps disingenuously) discussed.