The centre-half, who missed Saturday's 4-2 win over Crewe in League One due to a one-match ban, needs surgery to remove a floating bone and boss Steve Evans has ruled him out for the next few weeks.
Evans said: "He will be out for another two or three weeks as he has a small piece of floating bone removed.
"It is a minor procedure so it is in the short term he will be missing."
Morgan's absence means Evans is down to the bare bones in the central defensive position for Saturday's trip to Shrewsbury, with Miles Addison and Kari Arnason the only two players fit and available.
Meanwhile, the Millers look set to miss out on striker Nouha Dicko after the Wigan man discussed terms with League One rivals Wolves.
Dicko enjoyed a highly fruitful loan spell at the New York at the back end of 2013 and Evans hoped to bring him back on a permanent deal.
But Kenny Jackett's men look to have stolen in after he was at their game with Preston on Saturday.
Jackett said: "He was due here. I have not seen him but he was due here to speak to Kevin Thelwell (Wolves head of football development).
"I don't know whether we have agreed a fee to be honest.
"My concentration on the last day or so has been on the game.
"I can't say he wasn't here because he was due here to speak to Kevin about a move and that is the situation."