Opener Reddy made 196 after spending 576 minutes in the middle, while Vihari scored 191 during his 496-minute knock, as Hyderabad finished the third day on 423 for three in reply to Mumbai's 443 at the Rajiv Gandhi International Stadium.
While Reddy hit 27 fours and six while facing 426 balls, Vihari's 331-ball knock was laced with 18 hits to the boundary.
Resuming on overnight 138 for one, Reddy and Vihari put together a monumental partnership of 386 runs for the second wicket, which pushed the visitors onto the backfoot in the Group A match.
With seven wickets remaining, the hosts are expected to make the required 20 runs and take the crucial first-innings lead.
None of the Mumbai bowlers could make much of an impact, and struggled to counter the challenge posed by the duo of Reddy and Vihari.
Reddy and one-down bat, Vihari, were overnight 63 and 53, respectively. Both of them were out in similar fashion, though to different bowlers.
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Brief scores:
Mumbai: 443 all out in 126.3 overs
Hyderabad: 423 for three in 135 overs (Akshath Reddy 196, Hanuma Vihari 191).