The 20-year-old has been on their radar for two years after impressing scouts at a tournament in Mexico.
Germany U21 star Can is a powerful, left-footed player who is good on the ball.
He can also play in defensive positions. He joined Leverkusen from Bayern Munich last summer, and the Bundesliga winners have a first-option buy-back clause that could yet scupper any talks.
Liverpool have watched Leverkusen several times this season, with 18-year-old Julian Brandt also impressing.