By Rupam Jain and Rafael Nam

NEW DELHI/MUMBAI (Reuters) - As Indian Prime Minister Narendra Modi prepares to pick a new central bank governor, he will need to juggle competing demands from within his own party, the economy's need for more stimulus and investors' demands for an independent thinker.

Global and Indian investors are on edge as local media report that Modi could select the new governor of the Reserve Bank of India (RBI) as early as Tuesday (July 12).

Outgoing RBI Governor Raghuram Rajan stunned financial markets last month by announcing he would step down in September after a single three-year term to return to academia.

Rajan was held in high esteem for defusing a currency crisis, cutting interest rates and overhauling the way the RBI operates, including introducing inflation targeting. But his sudden exit after persistent criticism from Modi's right-wing allies has raised questions about the RBI's independence.

That has made the choice of the next RBI governor critical.

Of the candidates widely seen as on the short-list, three insiders -- current Deputy Governor Urjit Patel and former deputy governors Rakesh Mohan and Subir Gokarn -- would likely easily pass that muster, and are seen by markets as ensuring both policy continuity and the RBI's independence.

"The candidates from RBI (Gokarn, Mohan, Patel) would have been considered steady hands that would continue current Governor Raghuram Rajan’s inflation targeting framework (and be the least disruptive)," BNP said in an email to clients on Tuesday.

Less certain is State Bank of India <SBI.NS> Chair Arundhati Bhattacharya. She is a career commercial banker with no economics background, but may be seen as an ideal candidate with the inside know-how to best deal with $130 billion in non-performing loans weighing on the banking sector.

But Modi will also need to reconcile market demands with those from forces within his own administration.

As with previous Indian governments, Modi, along with Finance Minister Arun Jaitley, are bound to prefer somebody who is more accommodating on interest rates, especially given the new governor's term should run through the government's re-election in around three years.

Rajan was widely seen as hawkish on inflation, though the RBI has cut rates by a total of 150 basis points since last year. Recently he turned more stubborn on the prospects of further easing, complaining Indian banks were not passing along most of the RBI's rate reductions to companies and consumers.

Meanwhile, the Rashtriya Swayamsevak Sangh (RSS), the right-wing parent of Modi's Bharatiya Janata Party (BJP), is keen on a candidate who does not seek the limelight - an accusation often thrown at Rajan by some officials.

On Monday, media reported Modi policy adviser Arvind Panagariya was a potential surprise front-runner. His candidacy could raise uncertainties in markets because of his perceived closeness to the prime minister. He is also seen as close to the RSS and BJP President Amit Shah.

(Additional reporting by Neha Dasgupta in NEW DELHI; Editing by Kim Coghill)