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There are no direct connections from any of the major towns in Kerala to Hospet, which is the nearest railway station for Hampi.

From Kerala aim for one of the connected places for Hospet. Goa, Mangalore, Guntakal and even Bengaluru fall in this transit destinations.

By far train is the best possible option to travel to Hampi from Kerala. Though Hospet is the nearest railway station for Hampi, there are no direct trains (not yet at the time of writing this!) from Kerala that travel via Hospet.

But you have two broad choices. Travel to Goa and then travel to Hospet (and Hampi). Or travel to Guntakal Jn and then look for a connection to Hospet.

Of the two there are some half a dozen trains connect Kerala with Goa along the Konkan coast.

Cities in Kerala Thiruvananthapuram, Ermnakulam (Kochi/Cochin), Kollam, Alapuzha, Trissur, Kozhikode, Kannur etc are connected connected to Goa by Netravati express.

Daily trains like Kanyakumari-Bangalore Express connects Thiruvananthapuram with Bangalore via other cities in Kerala like Kollam, Kottayam, Ernakulam, Trissur, Palakkad.

To Guntakal there are only two trains (the Mumbai Express Train No:16332; and the daily Kanyakumari Mumbai Jayanti Express Train No:16382 ).
From Goa to Hampi opt for a train connection to Hospet ( Howrah Express , Train No:18048 ) or even there are overnight buses from Goa to Hospet.
While this may not be the shortest, you may also consider a third train option, that is via Bangalore. You can catch that overnight Hampi Express (Train number 16592) from Bangalore to Hospet.
By there way there is a 12 hour or so bus (Rajahamsa service by KSRTC ) connection from Mangalore to Hopset. If you are traveling from the northern part of Kerala, explore the feasibility of this option too.

See also other travel options to reach Hampi.