Alderweireld last week revealed that he wished to leave the Eredivisie, with Germany or England his most likely destinations.
That news was said to have alerted Liverpool and Tottenham, both of whom have reportedly shown an interest in signing the defender.
However, it now appears that Leverkusen, who finished third in the Bundesliga last season, are in pole position to capture Alderweireld.
"Bayer Leverkusen is the club who we are the furthest with in our discussions," Vic Alderweireld told Belgian paper Gazet van Antwerpen.
"We will quickly meet the club again."
Alderweireld is a ball-playing defender who is also capable of playing in central midfield.