The former Manchester United star is having his tonsils removed after suffering for some time with tonsillitis.
"I have been ill with tonsillitis for around ten weeks," he said.
"It's been a bit difficult for me recently.
"I've been suffering on and off and naturally you can't feel fully fit in that time.
"It's been very frustrating, I had hoped to play a few games this month then have my tonsils out."
Boro boss Steve McClaren confirmed Greening would be out for at least eight weeks, saying: "He'll be out for a minimum of two months after the operation."
Nash will also go under the knife, with his problem being a broken finger.
"He's having an operation on his finger and will be out for quite a while," said McClaren. "We're talking about months not weeks."
One plus point for McClaren is that Joseph-Desire Job is back from his serious injury and he confirmed that he would not be travelling to the African Cup of Nations, even if he received a late call-up from Cameroon.
"After a long-term injury going to the African tournament would have been too early," he revealed. "I prefer to stay with Middlesbrough.
"I was pleased with my comeback. My knee stood up to the test. I was not sure whether I would be a substitute, but the manager asked me to play from the kick-off."
McClaren was happy to welcome his African striker back.
"Job has been missing a long time. Early in the season Job and Malcolm Christie played well up front," he said.
"We lost both through injury and have struggled since. With Job fit again, it is like a new signing.
"He did remarkably well against Notts County bearing in mind he had only trained for two weeks. He had not even had a reserve game before his first-team comeback.
"He will be an asset in our packed programme."
Meanwhile, Gaizka Mendieta (virus) and Michael Ricketts (knee) are both serious doubts for the game with Fulham on Wednesday.
