Those at Anfield are reported to be plotting a summer move for a new goalkeeper, with their sights locked on the current No.1 at Stadio Olimpico.
Barcelona, meanwhile, are weighing up the merits of an approach for Turkey international winger Under.
Roma, though, have warned their European rivals off any bids during the next window, with club president James Pallotta telling Sirius XM: “It’s frustrating for me because people believe Roma is a supermarket.
“We do not want to sell players; these guys are on long-term contracts and we have no intention of them leaving the club.
“They have been incredible with us from day one and this season they have both been fantastic, they have big futures – just like Patrik Schick.
“Monchi faces the difficulty of financial fair play but we have identified a plan and strategy to continue to improve the team.”
Roma sporting director Monchi was the man to reveal Barca’s interest in 20-year-old talent Under.
"It's true that Barcelona have an interest in the player," he told Radyospor.
"Bayram Tutumlu [Under's agent] has a strong relationship with Barca and he informed me that there is definitely an interest there.
"We have not received any offers for the player but it would not surprise me if he was being tracked – he is a young, talented player who is scoring a lot of goals.”
Under, who only joined Roma from Istanbul Basaksehir in the summer of 2017, has netted six times in 24 appearances this season.
Club colleague Alisson has been starring at the opposite end of the field, keeping 17 clean sheets over the course of the season.
He also helped Brazil to shut out reigning world champions Germany in his latest outing during the international break.