The 23-year-old Greece international has made an outstanding start since arriving from Olympiacos during the summer, establishing himself as an important figure in Hull's return to the top flight.
Tzolakis became the Tigers' most expensive goalkeeper when they completed the deal, beating competition from several clubs across Europe. His performances have quickly justified the investment, with three clean sheets in his first six appearances helping Hull make an encouraging start to the campaign.
His opening performances were particularly impressive, as he became only the ninth goalkeeper in Premier League history to record a clean sheet in each of his first three matches in the competition.
That form has reportedly attracted Liverpool, who are assessing their options in goal amid uncertainty over Alisson Becker's longer-term future. Tottenham have also been linked with an interest in the Greek international.
According to reports in The Sun, Liverpool could explore a potential deal worth around £40 million, although no formal agreement has been reached. Hull City are understood to expect discussions with interested parties around the middle of November rather than immediately.
Tzolakis arrived at the MKM Stadium with a growing reputation after establishing himself at Olympiacos. He helped the Greek club win the Europa Conference League and subsequently gained valuable European experience against prominent opponents in the Champions League and Europa League.
His performances against Arsenal, Real Madrid, Barcelona, Ajax and Bayer Leverkusen helped demonstrate his ability to compete against elite attacking players. That experience made him an attractive target for a range of clubs before Hull won the race for his signature.
The goalkeeper has also impressed on international duty. His displays for Greece included a victory over Germany that ended Jürgen Klopp's unbeaten start as Germany manager, followed by a clean sheet in the return fixture. He also produced strong performances against the Netherlands.
For Hull, retaining Tzolakis would be a major boost as they look to consolidate their position in the Premier League. The club have ambitions beyond simply avoiding relegation, with their encouraging start raising the possibility of a push towards the European qualification places.
However, the growing interest in their goalkeeper could test Hull's resolve. A substantial offer would present a significant financial decision, particularly given the scale of their original investment.
Tzolakis remains focused on his responsibilities with Hull, and the club have no immediate reason to rush into negotiations. The expectation is that any serious discussions will take place next month, leaving the Tigers time to assess the situation.
With Liverpool and Tottenham reportedly monitoring his progress, Tzolakis' performances over the coming weeks could determine whether interest develops into a formal approach. Hull will hope to keep their standout goalkeeper as they pursue a successful return to the Premier League.