The 29-year-old Germany international has struggled for first-team oopportunities at the Emirates this season and has been the subject of interest from a number of clubs across Europe.
Inter made an initial offer last week that Gunners boss Arsene Wenger branded “a joke” before improving their offer on Friday.
Podolski landed in Milan to a mob of Italian supporters throwing Inter scarves around his neck, and he is expected to agree a deal to join the Serie A club.
Speaking after Arsenal’s New Year’s Day defeat at Southampton, Wenger said: “The first proposal was farcical. They came back a bit more serious, so we will see what happens now.
“In the transfer market it is not inevitable that somebody leaves – it’s very difficult to say.”
Podolski joined Arsenal from Cologne for £11m in 2012 and has scored 31 goals in 82 appearances in all competitions.
He featured fairly regularly over the past two seasons but has only started twice this term, and came off the bench to score an important late winner at Anderlecht in the Champions League.
Podolski is the third-most capped player in Germany’s history, and their third all-time top goalscorer, and was a member of World Cup winning squad in Brazil in 2014.
