The 18-year-old left back has forced himself into first-team proceedings under Ruben Amorim in the Primeira Liga over the last few months, which has caught the eye of clubs in the Premier League and in Serie A.
Mendes joined the club at the age of 10 and made his senior debut in June, going on to make nine appearances in the league as well as 14 for various age groups with the Portugal national team.
Portuguese newspaper A Bola noted the interest of both Arsenal and Liverpool earlier this week, and also mentioned Juventus and Milan as other clubs who are keeping a keen eye on the player.
They add that Mendes will only be allowed to leave if his buy-out clause is met, which is believed to be in the region of £41m.
That fee could be enough to force both clubs to take a step back in proceedings, with Klopp open about his side’s inability to match the spending of big-spenders Chelsea.
The former Borussia Dortmund boss said in an interview with BBC Sport earlier this month: "We live in a world at the moment with a lot of uncertainty.
"For some clubs it seems less important how uncertain the future is. Owned by countries, owned by oligarchs, that's the truth. We are a different kind of club.
"We got to the Champions League final two years ago, we won it last year. We became Premier League champions last season by being the club we are.
"We cannot change it overnight and say, 'Now we want to behave like Chelsea.'
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"They are signing a lot of players. That can be an advantage, but that means they have to fit together.
"It's about working on the training ground. That will probably be an advantage for us. We've worked quite a while with each other.
"You cannot bring in the 11 best players in the world and just hope a week later they play the best football."