Coric, who is seeded seventh in the draw, overcame experienced Spaniard Marcel Granollers 6-3 6-3 to move into round two.
Bedene beat another Croatian Mate Delic in straight sets on Monday but will have to be at the top of his game to see off 18-year-old Coric, who pulled off a surprise win against Andy Murray earlier this year and will have the home crowd in his favour.
"I have a lot of pressure here because everybody talks about me as a star," Coric told atpworldtour.com.
"Everybody expects me to win the tournament. This is something I have to deal with.
"Novak Djokovic and Roger Federer feel like this at every tournament."
Coric added: "I watched Bedene yesterday. He serves and moves really well. It is going to be a tough match."
Germany's Philipp Kohlschreiber is also through to round two after he beat Colombian Santiago Giraldo.
Top seed Gael Monfils will face Austria qualifier Bastian Trinker, who knocked out veteran Russian Mikhail Youzhny.