Pukstas has established himself as a key player for Hajduk Split in the past couple of seasons, contributing to the team with seven goals in 35 appearances.
The 19-year-old has earned comparisons to the likes of Thomas Muller due to his playing style and versatility.
According to the Sun, Arsenal scouts were present during the teenager's match against HNK Gorica last Saturday when Hajduk ran out 3-0 winners.
The report adds that The Gunners intend to continue to monitor his progress ahead of a possible transfer to North London next year.
Pukstas signed a new contract with Hajduk only last month that is due to run until 2027, but many expect that the talented teenager will be snapped up well before the end of his current deal.
The 19-year-old, who was born in Oklahoma, has represented the USA at the Under-20s level and also spent time at Barcelona's academy in the US before moving to Europe.
Arsenal have been known for keeping tabs on American players since Arsene Wenger was in charge and have signed the likes of Daniel Karbassiyoon, Gedion Zelalem, Matt Turner and Auston Trusty from the US over the years.