The 23-year-old Czech Republic international is currently at the World Cup with his national side after an impressive season in the Primeira Liga as Braga's first-choice goalkeeper.
According to the Daily Mail, Hull have submitted an offer of £12.5 million plus add-ons as an opening proposal for Hornicek, who stands at 1.97 metres and has established himself as one of the most promising young keepers in Portuguese football.
Braga are understood to want more than the offer currently on the table, and clubs including Leeds United, Inter Milan and Porto have also been monitoring Hornicek's situation in recent months.
Hornicek is said to be open to a move to Hull, but the fee remains the primary obstacle to finalising a deal.
The goalkeeper is under contract with Braga until the summer of 2028, which gives the Portuguese club leverage to hold out for a higher figure.
Hull have also cast their net wider in their search for a number one, making an enquiry for Bayern Munich backup goalkeeper Alexander Nubel.
The 29-year-old spent the last three seasons on loan at Stuttgart in the Bundesliga before returning to Bayern, and was on the bench for Germany's 7-1 opening World Cup win over Curacao.
However, Nubel is understood to be targeting a club capable of offering him regular first-team football at a higher level than newly promoted Hull can currently guarantee.
Beyond the goalkeeper position, Hull have also been exploring the Croatian market as they seek additional options.
The club would also welcome a conversation with Manchester United about midfielder Toby Collyer, who spent the end of last season on loan at Hull.
The 22-year-old made a positive impression during his time in East Yorkshire, and Hull's view is that a full pre-season under their system would allow Collyer to develop into a player capable of contributing immediately in the Premier League.
Hull's goalkeeping setup at the World Cup provides a useful illustration of where the club is heading. Ivor Pandur, who kept goal in the Championship season that earned promotion, is with Croatia as a backup, while Hull are now working to bring in a candidate to challenge him for the number one jersey at the top level.
For Hornicek, the move represents a significant step up from Portugal to the Premier League, albeit with a newly promoted club rather than an established top-flight side.
His age, contract length and World Cup involvement give him options, and Hull will need to move quickly and improve their offer if they are to see off the interest from clubs with more experience at the level they are joining.