The Deputy Prime Minister met Mr Modi at his official residence, the Panchavati in New Delhi, for the discussions about links between the two countries.
The Liberal Democrat leader, who is in India with a 40-strong trade delegation, discussed the prospect of bringing leading teachers from the UK's top institutions to teach in India.
Another idea the pair considered was the possibility of British experts helping to clean the Ganges, drawing on efforts to improve the water quality in the Thames.
Senior government figures have made a series of trips to India since the coalition came to power in 2010 in an effort to create a new "special relationship" and strengthen trade with the emerging economic powerhouse. Chancellor George Osborne and then Foreign Secretary William Hague visited in July, while David Cameron has made three trips to India as Prime Minister.