They closed the free transfer signing of Stephan Lichtsteiner and are pushing to close a move for centre-back Sokratis from Borussia Dortmund.
But one of their key targets is Leno, because the club does not believe Petr Cech is good enough as the first choice keeper anymore.
The player does have a £22 million release clause, which will make him easier to sign.
When speaking about the possibility of his exit, Leverkusen sporting director Rudi Voller is quoted by the Daily Mirror as saying: "Leno can leave but only on our terms.
"We are still in constant and open exchange with Bernd and his manager. Leno already had a concrete offer from Napoli in 2017, but we didn't have an equivalent backup at the time.
"He had to stay and he did really well last season, but we thought Bernd's desire to leave would resurface so and so with the arrival of Lukáš Hrádecký we have a high quality successor."