India lead the series 1-0 after thumping Sri Lanka by an innings and 144 runs in last week's second Test in Kanpur and have been given an added incentive to make it 2-0.
India are currently third on the world rankings behind South Africa (first) and Sri Lanka (second) but are close enough to leapfrog them both with a victory in Mumbai.
Climbing to the top spot would undoubtedly represent a major milestone in Indian cricket but captain Mahendra Singh Dhoni has tried to play down the significance, warning his players just to concentrate on the match and forget about the rankings.
"It will be an added responsibility," he said.
"Becoming the number one side is not important, we've got to maintain our performance level."
India's chances of beating the Sri Lankans have already suffered a blow with in-form opener Gautam Gambhir, the world's top-ranked batsman, pulling out of the match to attend his sister's wedding.
The left-handed Gambhir has played a significant role in India's great start to the series, scoring centuries in each of the first two Tests and sharing in a 233-run opening stand with fellow centurion Virender Sehwag in Kanpur.
He is expected to be replaced by Murali Vijay, an exciting but relatively inexperienced batsmen who played his only Test against Australia last year.
Batsmen have dominated the series so far with 10 centuries in the two Tests although Indian medium-pacer Shanthakumaran Sreesanth gave the seam bowlers some hope with a five wicket haul in the first innings in Kanpur.
Although they cannot win the series, the Sri Lankans are eyeing their own slice of history and are putting their faith in spinner Muttiah Muralitharan.
They never won a Test on Indian soil and the form of Muralitharan remains a real cause of concern with the world record holder managing just five wickets at a cost of 396 runs.
"We got the worst bowling conditions in the last two Tests," Muralitharan said.
"We didn't have the bowlers, that was one of the factors."
Sri Lanka captain Kumar Sangakkara remains confident Muralitharan can recover and play a starring role in what will be his final Test away from home.
"He is still the best bowler that has played the game," Sangakkara said.
"So you still have to have faith in him and trust him and keep backing him 100 per cent to do the job because he is going to do it at one point.
"We have still got a lot to achieve in Mumbai. Whether we can win the series or not, we can still equal the series and finish the series 1-1. That is a great motivation for us."