The Norwich frontman suffered the injury in a training ground friendly during the international break, ironically set up with the specific intention of giving him game time.
Becchio's loan spell runs until January and Evans has not ruled out the possibility of seeing him in a Millers shirt before then.
"He's had a CT scan which is as much detail as a consultant will want," Evans said. "We think it is going to be towards December if not into January before Luciano is back available to play football, which is a shame because we worked hard to get him in the type of shape we wanted him.
"We hope to see him in a Millers shirt if he is back available and fit because he is our player until the end of his loan spell.
"A lot of his rehabilitation will be done here after they make the decision what happens to him.
"There are two consultants having two current views at this moment in time, but there will be a senior consultant appointed by Norwich and he will make a decision w hether it is an operation or just time.
"Either way it will be not before Christmas time until we see him available.
"I think we will know in the next 24 hours whether he goes under the knife or whether he just has to rest it. The consultants are the experts and they make the decision."
Evans has been given consent by chairman Tony Stewart to bring in a loan replacement, though that will not happen before Leeds' visit to New York on Friday night.
"I have got that option," Evans added. "When we discussed the situation the morning after our closed-door game, the chairman gave me immediate consent to go back into the market.
"We have to sign better than what we have got and if we can do that we will.
"We have got Matt Derbyshire, Jordan Bowery, Paul Taylor, Jonson Clarke-Harris all vying for a place. That's if a play with two strikers of course.
"We have got some good strikers here, they are all good players."