Colombia is pro-free trade and signed an agreement with the United States last year.
However, its troubles have meant it is yet to attain the OECD status it is hoping for. It is likely Mr Key will reveal New Zealand's stance on that during his meeting with Mr Santos.
Colombia could also raise its wish to join Apec and possibly sign up to the Trans-Pacific Partnership.
Currently, neither country has a diplomatic base in the other.
Colombia's representative in Canberra looks after New Zealand, and New Zealand relies on its embassy in Chile to look after Colombian interests. But Colombia is expected to appoint a consul-general in New Zealand soon.
President Juan Manuel Santos, 61, has been in office since August 2010 and has made fiscal reforms and changes to allow Colombia to benefit more from revenue for oil and mining.
Colombia
*Fourth largest Latin American economy after Brazil, Mexico and Argentina.
*Population: 47 million
*Trade: New Zealand imports were worth $19.6 million from Colombia in 2012 - including $7.5 million of coffee. Other imports include cruise vessels, insecticides, ambergris. New Zealand exported gods worth $13.3 million last year.
*Visitors to NZ from Colombia in 2012 totalled 928.