The on-loan defender has played just eight minutes under Gerrard.
He has played just eight minutes since Gerrard was appointed in November and started six première ligue Jeux, all under former boss Dean Smith.
Gerrard said: “It’s very much in Ax’s hands and Manchester United’s hands. We want him here, we would be delighted to keep him. We don’t just see it as a loan until the end of the season.
“It’s something we want to explore further than that. We just hope we’ve got that opportunity.
“He’s a player we like very much and we’d like him to stay until the end of the season at least. Axel is his own man and will want to make his own decision.
“He is obviously owned by Manchester United which we all need to respect.I hope he wants to be part of this.
“In terms of what he has given on a daily basis, his leadership and training levels have been really strong and he is pushing to play.”
Gerrard suggested United have the opportunity to recall the 24-year-old in January but hopes it does not happen.
Il a dit: “Most loans, to give you a hint, normally have a recall option. I don’t think it makes sense to loan any player out in this day and age, especially when you are going through a pandemic, to not have a recall option. We will see how that one pans out in the short term. We want to keep him.”
Gerrard will return to the dugout at Brentford after missing Boxing Day’s 3-1 defeat to Chelsea having tested positive for Covid.
“I’ve had the worst Christmas on record but that’s not important,” added Gerrard.
“I’m very much involved in that result (against Chelsea) and the outcome of the game, that’s my responsibility. I was very much in touch with the staff and trying to get messages to the bench.
“It was me who selected the team and the substitutions within it, so it’s very much on my shoulders. I had a chance to rest in terms of being at home and being away from it but the work never stopped.
“From my point of view, I always want to be here. I want to be here and on the job itself. It’s nice to be back.”