Reid, in turn, is understood to be fuming with Mills' comments and there is the prospect now the defender may have played his last game for the club, despite only recently signing a new five-year contract.
"Danny is very professional in his outlook and has never had a problem with criticism, but he will react if he feels the criticism is unjust," said Featherby.
"He was obviously disappointed not to be involved on Sunday, particularly as he has been in the England squad for the last 18 months.
"His football means a lot to him. He loves the club and is desperate to do well and to be in the first team.
"I am sure Danny will have a meeting with Peter and I'm hopeful it can be resolved. In fact, I'm sure it can be.
"Danny's got a five-year contract and that's a long time to keep somebody out."
Featherby, a former athlete, insisted that Mills was fully prepared for pre-season.
"I know Danny's fitness levels as well as anyone, although it's none of my business what Leeds do," he said.
"They've first class people to say whether Danny is fit or not, but I do know he is fit because of the type of work I have done with him.
"He is as fit as any footballer and a number of top athletes I have worked with over the years."