According to the report in 'The Sunday Times', Andrew – the Duke of York and second son of Queen Elizabeth II – has known Modi for several years and met Modi at his London home last July, just days before his travel papers were granted.
Buckingham Palace refused to say what the two men had discussed but denied "categorically" that the duke had lobbied the UK government on Modi's behalf.
Modi's immigration status in the UK has provoked a storm in India after the UK-based newspaper disclosed that he rang up External Affairs Minister Sushma Swaraj to help him acquire his British travel papers.
Although she says she acted on "humanitarian" grounds, Swaraj has faced calls for her resignation over an apparent conflict of interest.
Modi, 49, travelled to London in 2010 after the Indian Premier League (IPL) became embroiled in allegations of match-fixing and illegal betting. He denies any wrongdoings and says he came to Britain because he faced death threats from criminals in India.
In March 2014, Modi was granted leave to remain in the UK after a court battle with the Home Office. He later applied for British travel papers, known as a certificate of travel.
In June 2014, Vaz wrote to Sarah Rapson, director-general of UK visas and immigration, asking if the papers "could be made available" to Modi, the paper reported earlier this month.
Vaz said he helped Modi because the former IPL chief had told him that his wife urgently needed cancer treatment abroad and that he needed to accompany her.
On July 2 last year, Vaz raised Modi's case with another Home Office official.
He wrote: "Mr Modi informs me that he met the Prince of Wales (Charles)... and he also met Prince Andrew two days ago. Both offered to help resolve this matter as he was complaining about missing the wedding of his sister and the inability to meet amongst others the president of the Seychelles."

Latest News from World News Desk