The 15 year-old midfielder made his Premier League debut in yesterday's 5-0 win against Leeds United.
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A second-half substitute, Dowman won a late penalty, converted by Viktor Gyokeres, during the victory.
Transfer expert Nicolo Schira is now reporting: "Excl. - Arsenal are planning to lock Max Dowman (born in 2009) with his first professional contract as soon as he will turn 16.
"Ready a deal until 2028 with the option. He is considered a future top player world class."
Dowman will turn 16 on December 31.