The German club, currently 14th in the Bundesliga, made no confirmation of the claim but Bild, which is generally well-informed, said he had accepted terms with the English Premier League side.
Podolski, 26, has agreed terms on a four-year deal that will see him earn an annual salary of some seven million euros ($9.2 million), the newspaper said.
The attacker was transferred in 2009 from Bayern Munich to Cologne, where he played at the start of his career.
The Gunners’ injured centre-back Per Mertesacker told German Sky Sports News on Wednesday that Podolski has been getting the low-down on life at the Premier League side through an exchange of text messages with him.
“He has been very interested in the last few weeks and sent me a text message or two,” Mertesacker said.
“I just responded to him positively, we have a good club here and everyone here has a positive opinion of ‘Poldi’.”
Podolski’s contract with Bundesliga strugglers Cologne does not expire until June 2013.
The player has made 95 appearances for Germany, is a veteran of two World Cups and was part of the team which reached the Euro 2008 final when the Germans lost to Spain.
Bild previously said that Cologne wanted 18 million euros for Podolski, who has scored 16 goals in 20 German league games this season — more than half of his team’s total tally of 31.