Leno was fortunate not to be sent off against Eintracht Frankfurt on Saturday and, had he been dismissed, it would have left the club with only inexperienced goalkeeper David Yelldell as cover.
Yelldell, 31, has yet to appear for the club in the Bundesliga, which is why a more experienced custodian to put pressure on the man Leverkusen bought from Stuttgart last season is at the top of their shopping list before the transfer window closes.
"We are considering things and Rensing is also in those thoughts," confirmed Leverkusen's director of sport Rudi Voller to the Bild newspaper.
Rensing has no future at second-division Cologne after being dropped in the summer.
Although he still has a year left on his contract, he is no longer training with the team and is instead keeping himself fit with a personal coach until he can find himself a new club with local rivals Leverkusen set to extend him an olive branch.