Braga president António Salvador confirmed the club had received a concrete offer for the 24-year-old, who turned it down.
"We even had one [a concrete approach]. The player himself, Hornicek, refused. There was a proposal from England, from the Premier League, but the player felt he shouldn't go to that club and we were happy for him to stay," Salvador told Portuguese media.
Hull City had matched Hornicek's £25.9 million release clause, while Benfica have also made enquiries over the goalkeeper in the past week. He returned to action in midweek, starring in Braga's 1-0 win away to Serbian side Železničar Pančevo in Europa Conference League qualifying.
Newcastle United's interest in Hornicek follows the collapse of their long-standing pursuit of Manchester City goalkeeper James Trafford, who is now set on a move to Leeds United worth around £45 million as he discusses inserting a release clause into his new contract.
Eddie Howe's side had also held a meeting over Brighton's Carl Rushworth, but reservations about his lack of top-flight experience have left Newcastle looking elsewhere. Rushworth remains a target for Coventry City, where he spent last season on loan, though the Championship club have already had bids worth £25 million rejected.
With Hornicek having turned down Hull and Newcastle now making contact, sources suggest a deal could move quickly should all parties wish to proceed, giving Eddie Howe a potential solution to his search for a long-term successor to Nick Pope between the posts.
Hornicek has made 88 appearances for Braga since establishing himself as their first-choice goalkeeper, keeping 24 clean sheets last season alone and earning a place in the senior Czech Republic squad. His performances have made him one of the more sought-after young goalkeepers in Europe this summer, with his release clause no barrier to interested clubs but his own reluctance to move the deciding factor so far.
Newcastle will hope their project proves more persuasive than Hull's, particularly if talks with Braga can be concluded without needing to negotiate above his existing release clause.