All that remains is for Arsenal to lodge their official bid - seen internally as a straightforward next step rather than a genuine hurdle - after which the 24-year-old winger would travel for a medical ahead of committing his long-term future to the club.
The chase for Tzolis gathered momentum this week once Leandro Trossard's exit to Besiktas opened up a spot in Arsenal's wide areas. Sporting director Andrea Berta has been the driving force behind the pursuit, having sent his assistant Maurizio Micheli to run the rule over the Greek in the flesh during a recent international fixture against Sweden.
It has been a productive campaign for Tzolis, who racked up more than 20 goal involvements in all competitions for Brugge across 38 games last season. English football is not entirely new territory for him either - he spent time at Norwich City earlier in his career, with loan spells at Twente and Fortuna Dusseldorf along the way before he settled in Belgium and rebuilt his reputation.
Crystal Palace tried and failed to prise him away 12 months ago, with Brugge knocking back their advances, and the Belgian side are said to have already begun lining up a successor.
Arsenal have been careful to frame the Tzolis business as independent of their hunt for a bigger-name forward.
Atletico Madrid's Julian Alvarez remains top of that list, despite competition from Barcelona and Real Madrid, while Aston Villa's Morgan Rogers is viewed internally as the preferred option for added versatility up front - talks over his signature are expected to intensify once his World Cup involvement with England draws to a close.
Should the Tzolis deal go through, it would mark Arsenal's third notable addition of the window, following the earlier arrivals of Piero Hincapie and Illan Meslier.