Vaz, chairman of the House of Commons Home Affairs Committee, has expressed serious concern about the deal because of demands by Indian information-technology giants, the Daily Express reported.
Negotiations over the EU-India Free Trade Agreement will re-commence when a top-level team from Brussels flies to New Delhi this week, the report said.
India wants to send skilled technology graduates to work in Europe, mostly in Britain, in return for granting London -access to huge banking and insurance markets.
Documents reportedly seen by the Daily Express revealed that Britain was willing to -accept 20,000 workers on intra-company transfer visas, which would bypass the government's -immigration cap.
The EU's chief negotiator Ignacio Garcia Bercero said talks over the deal had now reached a "critical phase".
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Prime Minister David Cameron, who is due to visit Delhi in February, made no mention of the Indian deal at the World Economic Forum in Davos, Switzerland, the report said.
Vaz said Downing Street needed to be more candid about the influx of migrants and the possibility that British graduates might be undercut and squeezed out of jobs.
"My concern is that although the government strongly believes in reducing the number of people who come into this country, this EU-India Free Trade Agreement comes with strings," Vaz said.
"And the strings are that thousands of visas will be given to Indian companies to enable them to come and work in the EU, including the UK, under the intra-company transfer arrangements."
"So it comes at a cost, and that will be the effect on real immigration. They say they want to bring down immigrant numbers into the tens of thousands, but that's not going to be achievable if you count all these intra-company transfer visas," the Indian-origin MP said.
The report said Western retail giants like Tesco and Carrefour want more market access in India but their moves sparked protests in September.
Bercero said other hurdles included Indian protests about EU dairy and poultry -exports.