The Magpies had identified a new first-choice goalkeeper as a priority during the summer and were also interested in Manchester City's James Trafford, Robin Risser, Zion Suzuki and Carl Rushworth.
Newcastle United ultimately moved for Hornicek, completing the deal shortly before Matthias Jaissle's arrival as manager.
Braga president Antonio Salvador confirmed the fee at the Portugal Football Summit.
“We believed Hornicek could stay this year, but he had a release clause and there was a club - Newcastle - who actually paid more. They paid €30.57m when the clause was €30m,” Salvador said, as reported by A Bola.
“It was a good deal for the club and for the player, who also had the ambition of playing in the English league.”
Hornicek had attracted interest from several major European clubs but was persuaded by Newcastle's plans for him.
The Czech goalkeeper said Newcastle were the final club to enter the race but quickly made their intentions clear.
“When I saw it on social media, I was surprised myself," he said. "I had no idea about the interest of such clubs. It was more about speculation.
“I’ve had several offers. Newcastle actually joined last, but they immediately outlined the project to me. They told me what they expected from me and promised the role of No 1. Plus, they acted very quickly, and there was no problem even with the money Braga wanted for me. I didn’t hesitate for a minute.”
Hornicek has made six appearances for Newcastle since arriving, keeping clean sheets against Tottenham and Millwall while making several important saves.
The early performances have quickly justified Newcastle's decision to pay slightly more than his release clause.