“A supporter jumped over something, slipped and kicked his ankle. It is swollen but we have to see,” Klopp said after Adrian saved Tammy Abraham’s penalty in the shootout win.
The incident happened as the Liverpool players were huddled together celebrating the match-winning save.
A supporter, attempting to join the group, ran towards them, slipping as he was grabbed by a steward, and collided with several players.
Klopp said: “It’s crazy. We love our fans, no doubt, but if they could stop doing that. We played against City and someone ran on the pitch then.
“At Norwich, there was someone as well.
“It’s not funny. Like the girl in the Champions League final, she made money. What can we do?”
Klopp says Liverpool’s third-choice keeper Andy Lonergan, 35, who was an unused substitute for the game in Istanbul which finished 2-2 after extra time, is “ready” to deputise for Adrian if the 32-year-old is ruled out.
“Andy Lonergan is in training with (Caoimhin) Kelleher and we have them if we have to make a decision,” Klopp said of his two back-up options.
“Adrian was a week before, Lonergan has trained with us. We will see. Kelleher has potential but is not 100% yet.”
It comes after Roma indicated they were willing to make a €20m deal to sign the Croatia centre-back only to fail to make an official offer.
Instead, Roma are understood to have used intermediaries to attempt loan Lovren for a small fee.