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Regular number one Alisson suffered a shoulder injury last week and Adrian took his place in the starting lineup against Aston Villa.
The Reds would go on to lose the game 7-2 and the Spaniard made a number of mistakes at Villa Park.
Alisson is set to be out for around six weeks and after last Sunday's disastrous display from Adrian, The Mirror have reported manager Jurgen Klopp has his eyes on Butland.
While the international transfer window closed on Monday, the domestic deadline isn't until October 16.
Former England international Butland is one of the highest earners at Stoke, but hasn't played for the club in almost three months.
He made his Three Lions debut as a 19-year-old and was expected to become one of the world's top goalkeepers.
Following Stoke's relegation from the Premier League in 2018, the club slapped a £30 million price-tag on Butland but nobody met their valuation.
Gazzaniga hasn't played for Spurs so far this season with captain Hugo Lloris established number one.
Summer arrival Joe Hart has been preferred for the club's Europa League campaign meaning the Argentine is firmly third choice.
Despite Adrian's performance against Villa, Klopp was quick to point out the defence let him down.
"Not only because of tonight, it's now easy and people are like this - I'm not sure if journalists are like this as well - but now people will go for Adrian and stuff like this," he stated per Metro.
"Yes, the first goal was not cool, of course. But apart from that, I don't think he had anything to do with all the other goals pretty much.
"We didn't help him tonight, let me say it like this, or we even did the opposite. He's a really good goalie who played 11 [Premier League] games for us last year and I think we won pretty much all of them.
"It's all fine. Our goalie was not the problem. Only around the first goal. And the reaction for that first goal was not his problem, that was then our problem. We have to do that better, we can do that better."