Dowman, 16, set two records last season as the youngest player to score in Premier League history and for Arsenal. He has yet to feature in the top flight this campaign, but marked his first appearance of the season with two goals in Arsenal's 4-2 Carabao Cup win at Ipswich on Tuesday.
Manager Mikel Arteta stressed he remains keen to keep the youngster involved with his senior squad, but said the club would continue to assess what is best for him.
"We will evaluate every six months where we are and what is the best thing for him to do and then make the right decision," Arteta said.
"It depends on a lot of factors. What he needs, and what the team needs as well, the availability we have in the squad, the minutes that he is playing and the qualities of those minutes. We have to be open, but now the focus is to be here and to impact the team."
Dowman, who completed his GCSEs over the summer, becomes eligible for a loan move when he turns 17 on 31 December, with the winter transfer window opening the following day. Luton Town manager Jack Wilshere, the former Arsenal midfielder, is believed to have expressed interest in taking the youngster to League One on loan.
"We haven't really listened to any of the possibilities because we were crystal clear that we wanted to have Max in the squad, that we wanted him to be part of the pre-season and then start to see where we are from there," Arteta added.
Dowman featured 13 times for Arsenal last season as they ended a 22-year wait to win the Premier League, and Arteta was careful not to rule out a loan while also stressing the risks of moving too soon.
"There have been many cases of that age when they have gone out and been successful and others not," he said. "In my particular case, I was 17 when I went to Paris [Saint-Germain]. It depends on the atmosphere that you find there, and how much the coach believes in you as to whether you go there just to be part of a squad or play minutes.
"Which country, it depends. But now he is in a good place, let's keep him here."
Arsenal have won all seven of their domestic and European fixtures this season, starting with victory over Manchester City in the Community Shield, and Dowman will hope his performance against Ipswich puts him into contention for Saturday's trip to Brighton.
"The best way to knock on the door is to do it on the pitch, not in the office," Arteta said. "When a player delivers constantly on the pitch, every single day in training, that's the best way, that's the best signal for the manager. That's the best signal for his team-mates, too.
"And what Max did the other day, alongside all the other players as well that haven't played many minutes this season, was really impressive."