The Hammers suffered a 2-1 defeat to Stoke away from home at the Britannia Stadium in the Premier League.
Former Nottingham Forest winger Michail Antonio put the London outfit ahead midway through the first half, but second-half goals from Giannelli Imbula and Mame Biram Diouf saw the Potters launch a successful comeback.
The defeat means that West Ham finish seventh in the Premier League, but they still have a chance of playing in the Europa League next season.
If sixth-placed Manchester United, who have a game in hand, win the FA Cup final against Crystal Palace next weekend, then the Hammers will feature in the prestigious European competition.
Terzic has admitted his disappointment at West Ham’s loss to Stoke, but he has urged the fans to remain optimistic about Europa League qualification.
Disappointing day in a great year full of memories. Europe was just inches away. It's not over!Thx for everything!⚒ pic.twitter.com/8yhxebZciu— Edin Terzic (@edin_terzic) May 15, 2016
To the Stoke fans.— Don Hutchison (@donhutch4) May 15, 2016
My honest opinion is West Ham battered you.
You dug in 2nd half.
Given/S,cross and my Stoke MOM Imbula