The 17-year-old forward joined the Serie A giants last month after emerging as the jewel in the crown at Hammarby, the Swedish club which is co-owned by Ibrahimovic.
Roback impressed on his debut last week, scoring in 4-2 win over Novara soon after completing his €1.5m transfer.
As well as focusing on bolstering Mikel Arteta’s first team roster, Arsenal have concentrated on adding to their Under-23 squad this summer and have already recruited the likes of Jonathan Dinzeyi, Tim Akinola, George Lewis and Anass Salah-Eddine.
The Gunners thought they had a deal in place to sign Roback, but the influence of Ibrahimovic, who made a triumphant return to San Siro last season, scuppered Arsenal’s plans.
‘Milan were tracking me, but I was all ready for a transfer to England and Arsenal, because I realised they really wanted to invest in me,’ Roback told Swedish newspaper Expressen.
‘Bayern Munich were interested too, but in the end I followed my instinct and chose Milan, even if I knew Arsenal already had a deal in place with Hammarby.
‘I had my first senior team training session on September 1 and I was not anxious, not trembling, I felt confident. This club paid a pretty important price for me, so I can’t go around thinking ‘oh wow, I’m training with the first team.’ It was important for me to prove myself.
‘After the session, (coach Stefano) Pioli told me I’d done really well and praised me to some of the other players, too.’