KOLKATA (AFP) - Millions of voters headed to the polls on Monday (April 4) in two Indian states, with Prime Minister Narendra Modi's party facing a tough fight as it tries to tighten its grip on power nationally.

Mr Modi's Bharatiya Janata Party (BJP) must win state elections to gain more seats in the nation's Upper House of Parliament, which has been blocking reforms seen as crucial to fuelling economic growth.

Most members of the upper house, which has obstructed measures such as a planned standardised goods and services tax, are indirectly elected by state legislatures.

The Hindu nationalist BJP is seen as having little chance in the large rural state of West Bengal in eastern India against a feisty chief minister popular with millions of impoverished voters.

It has a stronger chance of seizing power for the first time in the tea-growing state of Assam in the north-east, where it has promised to crack down on illegal immigration from neighbouring Bangladesh.

Mr Modi's party swept to power in a general election two years ago promising business-friendly reforms to overhaul the economy, but lost out in two critical state polls in 2015.

Analyst Neelanjan Sircar said the ruling party desperately needed a win in state polls this year.

"The BJP is clearly not doing well in the state elections and if they do not win one in 2016, they would have gone without having won a single state election for nearly two years, which is not good for any party," said Mr Sircar, a senior fellow at the New Delhi-based Centre for Policy Research.

"The only state in which the BJP may do well is Assam, and it is important for them to win this so that their base feels energised and the morale of the party workers is boosted."

Polls opened at 7am on Monday, with some 3.8 million voters eligible to cast their ballots in West Bengal and another 9.4 million in Assam.

Elections in both states are being held in phases, with around 85 million people eligible to vote in total.

Security was tight in West Bengal, with several of the contested seats located in impoverished regions where Maoist rebels have long been battling government rule.

Armed police have been deployed, along with helicopters, to try to ensure polling runs smoothly, election commission officials have said.

Chief Minister Mamata Banerjee, a 61-year-old former national railways minister known affectionately as "didi", or elder sister, is expected to remain in power.

But a row over the collapse last week of a flyover under construction in the state capital Kolkata, which claimed 26 lives, could cost her Trinamool Congress party some votes.

Ms Banerjee's political rivals have accused the government of failing to tackle problems with the project that began in 2009 and was only supposed to take 18 months.

Mr Biswanath Chakraborty, a Kolkata-based political scientist, said the disaster had "tainted the party for the first time since it came to power five years ago".

In Assam, known for its tea plantations and a myriad of rebel insurgencies, the BJP has teamed up with local parties that support indigenous rights and has pledged a crackdown on illegal immigration from Bangladesh.

Migrants have long been accused of illegally entering the state from Bangladesh and grabbing land, causing tensions with local people and sporadic outbreaks of communal violence.

Analysts say the popularity of the traditionally dominant Congress party is waning in Assam, offering the BJP its one chance of state election success.

The southern states of Tamil Nadu, Kerala and Pondicherry will hold elections later this month and next, but the BJP is not expected to win in any of these.

Counting and the release of results for all five states will take place on May 19.