The 30-year-old has taken plaudits for his fine form during Luton's recent run of one defeat in their past five games and Edwards said the benefit of a settled spell in the side is paying dividends.
"I am not going to give you a headline," said Edwards. "It is not for me to say. I am just happy that he is playing really well for Luton.
"That is all any player who wants to get selected for their national team can do."
Barkley has 33 caps and six goals for England but has not featured for his country since scoring twice and assisting once in a 6-0 win in Bulgaria in October 2019. That game was overshadowed by racism from the crowd.
So far this season, he has thee goals and three assists from 21 appearances for Edwards' side.
"We love Ross and are delighted with him," Edwards said. "He has got a home here with a group of players, mates, staff and a football club that support him and love him for what he is really good at.
"He is hungry and wants to show people how good he is. He had a couple of difficult years by his really high standards, but all he has shown [since joining Luton] is how good a player he is."