Negotiations between the two nations, which have been taking place during Christmas, will continue into the New Year.
Prime Minister Mr Sunak and India’s premier Narendra Modi, below, are said to be keen to get the deal wrapped up by April.
The mega tie-up could dramatically boost trade between Britain and India which is currently worth £36billion.
Set to become the world’s third-largest economy, India would be the biggest country yet to strike a free trade agreement with the UK since the nation left the EU’s trading system in 2021.
Britain hopes the pact will open up trade in whisky and cars to India, as well as services and investment opportunities.
Meanwhile, New Delhi wants better access for manufactured goods, services and a deal on work visas.
Mr Sunak had hoped to get an agreement in place in the autumn. But lack of progress in opening up Indian markets to British professional services, including law and accountancy firms, meant the talks stalled.
Tariffs on electric vehicles have also proved a stumbling block.
It is hoped a deal can be signed and sealed before India’s general elections begin on April 1.
A source close to the trade talks said: “We have made a lot of progress but the last stuff to do is the hardest. We have negotiators out there most weeks going through the details and we have a deadline of their elections.
“Both Rishi Sunak and Modi remain keen, so it’s just a case of seeing if we can get it over the line.”
A 14th round of talks is due to take place in January. Liz Truss was widely criticised as International Trade Secretary for rushing a trade deal with Australia in 2021, so it could be signed ahead of a G7 summit.
Her successor, Kemi Badenoch, is keen to avoid the same mistake. A Government source said: “Kemi is clear we won’t settle for anything that isn’t a good deal.
“She wants this to be welcomed by the various industries who stand to benefit, so no half-measures.”
UK-India trade talks began in January 2022 and Boris Johnson set a deadline to get a deal done by that October. But he was ousted as PM in July and the deadline came and went. Ms Badenoch has already signed a trade pact with US state Florida. Meanwhile, talks with Turkey are to begin shortly.
Britain is also eyeing a trade and security deal with Morocco, which could open up huge opportunities from the entire African continent.
The UK is also joining the Comprehensive and Progressive Agreement for Trans-Pacific Partnership trade bloc.
