Prime Minister Narendra Modi's Bharatiya Janata Party denies any wrongdoing by External Affairs Minister Sushma Swaraj or the top elected leader of Rajasthan state, Vasundhararaje Scindia.
The scandal, which relates to the investigation into the founder of India's Twenty20 cricket tournament, is the first to hit Modi's government since it came to power last year.
The upper house of Parliament was adjourned without conducting any business Tuesday, the opening day of its three-week monsoon session, with the government rejecting the resignation demand by the Congress party and others.
"There can be no resignation. If you want discussion, we can have it," said Parliamentary Affairs Minister Mukhtar Abbas Naqvi.
That remark infuriated Congress party lawmakers, who charged into the prohibited area around the chairman's seat and shouted, "Corrupt officials resign, then we will debate."
The confrontation over the issue is expected to delay the passage of key legislative proposals crucial to Modi's economic agenda, including a land acquisition bill and a new sales tax bill.
The Indian Premier League Twenty20 cricket tournament was founded in 2008 by Lalit Modi, who is not related to the prime minister, but it was soon dogged by allegations of corporate corruption, money-laundering and tax evasion, as well as secret deals to hide the teams' real owners.
Lalit Modi was removed as IPL chairman in 2010 and later banned for life from holding any positions in the sport by the powerful Board of Control for Cricket in India.
He denied allegations of financial wrongdoing and moved to London in 2010 as Indian authorities launched an investigation against him.
The current scandal began six weeks ago, when Britain's Sunday Times reported that Indian-born British Labor lawmaker Keith Vaz had used Swaraj's name to put pressure on the top British immigration official to grant British travel papers to Lalit Modi.
Swaraj tweeted that she told Britain's High Commissioner to India, James Bevan, and Vaz that granting Lalit Modi the documents would not harm the two nations' relations.
Lalit Modi and his attorney said that Scindia had given an affidavit to British authorities in 2011 supporting Modi's immigration application seeking resident status in the United Kingdom.
Scindia's son, Dushyant Singh, a lawmaker, received 113.6 million rupees ($1.85 million) from Lalit Modi in 2008 as investment in his company.