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.
He said: “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."