The Bundesliga side admit that a transfer for the striker is not financially viable, while the club's president claims his potential return could upset the team's uniformity
The 29-year-old's only start for the Gunners this season was in September's 2-1 defeat to Southampton in the League Cup and the Germany international has revealed that he will seek clarification over his future.
Arsene Wenger insists that he has no plans to sell Podolski during the winter transfer window, while Schalke claim that they have no intention of bidding for the forward.
Podolski made over 160 appearances for Koln during two spells with the club but the German side, who sit 10th in the Bundesliga, admit that a move is not financially viable.
"We do not have €5 million and we can't pay the salary for a player like Podolski," their managing director, Jorg Schmadtke, told Bild.
"Podolski needs a team where he can play to its strengths and bring them to full advantage. We can not currently offer that. I told him our point of view and I think Lukas sees it the same way."
The German club's president, Werner Spinner, also believes that Podolski's potential return could have a negative impact on the team, as well as the player.
"It would be difficult to integrate a 121-cap international, who also plays in the Champions League with Arsenal," he said. "We will never again impose such a responsibility on one player.
"No one has anything against Podolski returning to Koln - he is a popular figure - but we do not want to jeopardise the team's uniformity."