Liverpool are reportedly in the market for a new goalkeeper before the window closes, after first-choice stopper Alisson Becker suffered a shoulder injury in training ahead of the Reds' 7-2 loss to Aston Villa.
Adrian deputised for the Brazilian but was at fault for Villa's first goal, giving the ball away inside the six-yard box to gift Ollie Watkins the first of his three goals.
The Argentinian performed admirably last season when he stepped in to replace the injured Hugo Lloris for 18 Premier League games.
Alisson is expected to be out of action for up to six weeks, with Klopp telling Sky Sports before the Villa loss: "Yesterday an incident in training, very unlucky — two players clash with each other, one gets up, one doesn't.
"The one who didn't get up was Ali. So, after the first few things, we don't know 100 per cent. But he cannot play today.
"We have to do further assessment; but he's injured, let me say it like this. Now we hope it will go really quickly. It's an international break, I'm pretty sure after that he will not be ready. Then we have to see week by week.
Speaking after the game, Klopp added: "We were slightly fortunate but it is not that he can play against Everton. I think six weeks is maybe more likely but I don't know. So we have to further assess this thing and that is how it is."
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Liverpool legend Jamie Carragher did claim the club could look to sign a goalkeeper on deadline day, saying while on commentary for Sky: "Liverpool actually have a pretty decent win ratio with him in the side. It's just obviously the type of goals he lets in. It's a worry.
"It wouldn't shock me if Liverpool didn't try to do something on Monday. A new keeper might not have been in their thinking [before Sunday]. It could be now...."