Cardiff had lodged a club-record bid for the 20-year-old, but Hull City believe the chance to offer Premier League football gives them the edge in the race, with Tromso holding out for a fee in excess of £8m.
The 6ft 4in defensive midfielder has captained Norway at youth level and is a current Under-21 international, having scored seven goals and added two assists in just 14 league appearances for Tromso this season, form that has also drawn interest from RB Salzburg and Turkish side Besiktas.
Rangers were the first English-speaking club to move for Hjerto-Dahl, having a bid of around £5.3m turned down earlier in the window, but their interest has since cooled amid the growing competition, with the Ibrox club reportedly unwilling to match the rising asking price.
Cardiff, long-term admirers of the midfielder, have also seen a bid in the region of £7.8m rebuffed, leaving Hull as the club now best placed to complete a deal, helped considerably by their Premier League status following promotion.
Hull are also pushing forward in parallel talks for Nice winger Mohamed-Ali Cho, as the newly promoted club looks to strengthen multiple areas of the squad ahead of what is expected to be a difficult first season back in the top flight.
Securing Hjerto-Dahl would represent a statement of intent from Hull, whose recruitment team have identified the Norwegian as an ideal long-term option in central midfield given his physical profile and leadership qualities at youth level.
Tromso head coach Jorgen Vik has been repeatedly quizzed on the futures of both Hjerto-Dahl and team-mate Tobias Guddal amid the transfer speculation, and has praised the pair's mentality throughout the process, describing them as "very mentally strong" in the face of constant interest.
With Tromso still competing in Europa League qualifying, any transfer is unlikely to be rushed through mid-competition, giving Hull time to press home their advantage over their Championship and Scottish rivals before the window closes.