The veteran full back damaged knee ligaments during training last Friday and is set to be sidelined beyond the lifetime of his current deal.
Richardson signed a contract until December 30 earlier this month, extending his initial short term deal with the club.
But he will not feature for the club before the end of the deal following his injury.
"There was about 20 seconds left on Friday's session and he's reached for the ball, his foot has planted and he's done his knee ligaments," boss Darren Ferguson said.
"He certainly won't be available before his contract runs out. He's contracted until December 30 and he'll be injured until that.
"I feel for him. He's a great lad, I've said it many times.
"He just keeps getting injured and this one was an unlucky one - the last seconds of the session.
"He was in a lot of pain.
"The scan showed it was ligament damage. I was worried it wasn't going to be his cruciate which fortunately it's not.
"But you're definitely looking at a couple of months."
Richardson has managed just six appearances for Rovers since joining in August.
His only full 90 minutes came in the FA Cup defeat at Oldham Athletic earlier this month and he last featured in the Checkatrade Trophy clash with Port Vale last week.