Tsygankov and Gutierrez were both prominent figures in the Girona side that defied expectations to finish in third place in La Liga last season.
The Ukrainian winger registered eight goals and seven assists in 30 league appearances last term, while contributing with two goals and nine assists in 35 La Liga matches.
Tsygankov and Gutierrez both remained at Estadi Montilivi in the summer, but a number of their teammates were prised away from the Catalan club, including Artem Dovbyk, Aleix Garcia and Yan Couto.
Girona may have reason to be concerned that Tsygankov and Gutierrez could attract offers in future transfer windows, with two Premier League giants believed to be keeping tabs on their progress.
Arsenal, Liverpool monitoring Girona duo
According to Caught Offside, Arsenal and Liverpool both sent scouts to watch the pair in Girona's Champions League meeting with PSG at the Parc des Princes.
The report suggests that Arsenal boss Mikel Arteta and Liverpool manager Arne Slot are both admirers of the Girona pair.
In addition to Arsenal and Liverpool, representatives from Aston Villa were also said to be watching Tsygankov and Gutierrez in the French capital.
Those in attendance for Wednesday's fixture would have seen Girona frustrate PSG until a Paulo Gazzaniga error handed the hosts an 89th-minute winner.
Gutierrez played the duration of the contest on the left side of the Girona defence, while Tsygankov played 88 minutes before he was replaced by Alejandro Frances.
How much will Tsygankov and Gutierrez cost?
Although Girona offloaded several key players in the summer, they would reportedly be open to selling Tsygankov and Gutierrez if they receive enticing offers in the January transfer window.
There is a belief that the attacker and former Real Madrid defender will command a combined fee of €70m (£58.9m) to €75m (£63.1m).
Tsygankov and Gutierrez are both said to be keen on a Premier League move and their input could be crucial in whether they leave in January or remain until at least the end of the season.
The winger has played in Ukraine and Spain in his club career, while the former Spain Under-21 international has spent his entire career in his homeland.