The 16-year-old is yet to make a senior appearance this campaign after being an unused substitute four times, including during last Wednesday's Champions League win at Napoli.
Dowman will hope to change that when Arsenal travel to Ipswich for Tuesday's Carabao Cup fourth-round tie, although Arteta stopped short of guaranteeing him a starting place.
The youngster has gained further experience with the Under-21s and featured in their 4-1 defeat by Chelsea at the weekend, while Arsenal's squad depth has made opportunities harder to come by.
Arteta was able to name Noni Madueke, Eberechi Eze and Viktor Gyokeres among his substitutes for Saturday's 2-0 win at Sunderland, with Premier League rules limiting matchday squads to 20 players.
"Max will have his chance. He will have his minutes and he will have to earn them like anybody else," Arteta said.
"You know the talent that we have, the players that we have there, we need to help him and find a way to make sure he continues to develop."
Dowman made 13 appearances last season and became the youngest player to start a Premier League match against Crystal Palace. He later became the competition's youngest scorer when he netted against Everton.
Arsenal head into Tuesday's game after a demanding run that has included trips to Napoli and Sunderland, with Bukayo Saka acknowledging the strain on the squad.
"It's been a massive week for us. There has been a lot of travelling. People don't understand behind the scenes how it's been, but I'm really happy with how we've managed this," Saka told the club website.
"It's amazing for us, so credit to the team, the staff, all of us. We need to keep going - it doesn't stop."
Despite the Carabao Cup being behind the Premier League and Champions League in Arsenal's priorities, Arteta expects his players to be fully motivated.
"It's the next game ahead and the next competition," he said.
"We will look at the players' eyes, they will say they want to play, they want to win the game, they want to improve and they want to review the (Sunderland) game and they will be willing for me, because I know the boys."
Arsenal, who lost last season's final to Manchester City, will be looking to maintain their strong start to the campaign at Ipswich.
Jamie Carragher believes Mikel Arteta is becoming the Premier League's equivalent of Sir Alex Ferguson after the Arsenal manager launched a strong public criticism of the officials following the victory over Sunderland.
Patrick Vieira has confirmed his commitment to Senegal after expressing his pride at taking charge of the national team ahead of the 2027 Africa Cup of Nations qualifiers.
Matt Donohue and Blake Antrobus will not officiate in the Premier League this weekend after the video officials were held responsible for an error in Erling Haaland's winning goal in the Manchester derby.
Arsenal are set to seek clarification from Pro Ref over the penalty decision that went against Ezri Konsa during Saturday's victory over Sunderland.
