Newcastle are keen on Barkley with Luton relegated.
Edwards said: "Ross has been a really important player for us and that is what we want (to keep him).
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"We have had a good chat. We want to be open and honest with each other.
"He is an ambitious person, as we all are. He recognises what this club has done for him and what we have done for each other. He has loved this season at the club.
"Nothing is off the table and we are really hopeful he will stay but we are also realistic in this day and age."