The Englishman decided to move on from the relegated minnows, who will be in the Championship next season.
Barkley, who has signed for Aston Villa, shone for Luton in the Premier League but could not keep them in the division.
'We are disappointed to see Ross go, but we completely understand it and he leaves with our best wishes and thanks," Edwards said of the departed star.
"It was great to work with him for a season. He was amazing, provided lots of good memories for us all and the supporters fell in love with him, but I’m sure everyone will understand the opportunity.
“Playing in the Premier League is important for Ross, as is Champions League football. He still wants to play for England and believes this (move to Villa) gives him the best chance to do so. We’ve had a good chat and Ross has thanked me and the club. We are parting on good terms and you never know what the future holds, I’ve got to say that.
“He loves all of the supporters and he wants to come back to watch a game. He actually said, ‘I want to come back for the Watford derby!’ So let’s give him a great reception when he does come back during the season. He’ll always be welcome.”