The 24-year-old is on a season-long loan with the Potters from Paris Saint-Germain after his chances of regular first-team football at the Parc des Princes were diminished by the signings of Neymar and Kylian Mbappe.
The Spaniard spent the second half of last season on loan with Las Palmas and there is a sense that his time with the French club may be over.
He has enjoyed a decent start to life at the bet365 Stadium, scoring the only goal of the game in a 1-0 win over Arsenal on his debut.
He now hopes to be part of a successful campaign that could see Mark Hughes' side back in Europe after they last played in the Europa League in the 2011-12 season.
"My goal is to help Stoke enter Europe," he said. "For this we have to continue working as we have done, this way the victories will arrive. If I can have a good year then I’m sure that big things will happen."
Jese made his name at Real Madrid and while playing for Stoke may not be as high profile as life at the Bernabeu, the Gran Canaria-born forward is enjoying the experience and feels he will grow as a player in England.
"(I have adapted) much better than expected, really," he added. "The club and my team-mates have been great since my arrival and are always available to help me in whatever I need. I really am happy with the decision to move here.
"The key was trust placed by Mark Hughes on me. I was attracted to his ambition, he wants to make Stoke one of the most important teams in England.
"I had wanted to experience playing in England for a long time. In the Premier League I hope to recover my best form."