The Ivorian central defender was a frequent starter for Xabi Alonso's side last season, helping Leverkusen to a sixth-placed finish and a run to the Europa League semi-finals.
Kossounou made 35 appearances for the club last season, and has become a regular for the Ivory Coast national team.
The Daily Mail claims that Palace are interested in an initial loan with an obligation to buy for the 23-year-old defender.
Despite missing the final few weeks of last season with injury, Kossounou has since returned and featured three times for Leverkusen in pre-season.
Palace's transfer activity has been quite slow so far, with only Jefferson Lerma coming in at Selhurst Park with the season just a week away.
Roy Hodgson is struggling in some areas, with defence being one of them, as they could start the season without a recognised left-back.
The centre is also cause for concern given the supposed interest from other Premier League clubs in Marc Guehi.
Kossounou is a centre-back by trade, but can also fill in as a right-back on occasion, so his versatility could prove useful given Palace's lack of options at present.
Nathaniel Clyne and Joel Ward have been Hodgson's only options at full-back so far, so adding more strength-in-depth will be imperative.
Hodgson hinted earlier this week that improvements still need to be made because the squad is not as strong as last season, following the departure of five first-team players.
In the aftermath of their friendly with Sevilla, Hodgson said: "There's a bit of work for the chairman and sporting director to do between now and the end of August, because the team we put out today was in our opinion, at this moment in time, our best team, but we don't have the strength behind that team that we had last year."
A loan move for Lewis Hall from Chelsea has also been touted, which could help solve their current issues at left-back.
The club also hopes to have a move for Matheus Franca from Flamengo completed ahead of the new season.
Palace start their Premier League campaign with a trip to newly-promoted Sheffield United at Bramall Lane next Saturday.