Greenwood was arrested on Sunday for alleged rape and assault on a lady. He has since been suspended from first team activities.
Lingard was tipped to leave the Red Devils last month due to a lack of game time and he has less than six months on his current contract.
Lingard enjoyed a productive six-month loan spell at West Ham United in the second half of last season. The Hammers were once again interested in his services while Newcastle United were also keen on a move.
"Three weeks ago, Jesse didn't want to leave, then he changed his mind," Rangnick told reporters. "I had a chat with him 10 days ago and could fully understand why he wanted to leave for game time, for his future and also for the World Cup.
"I said in case he found a club he wanted to play for and also found a solution with our club, I would allow him to leave. But obviously with all the developments in the last [few] days, things changed a little, and we also had no agreement with any other club.
"So at one stage we had to take a final decision with the board, and we decided to keep Jesse until the end of the season."
When pressed further as to whether Greenwood's arrest had been a factor, Rangnick added: "In a way, yes, but in the end the board told me they couldn't find an agreement with those clubs interested in him, so with the window closing on Monday evening, in the afternoon the board said they would rather he stayed.
"For me it was a decision I could fully understand and accept. It was two things, we had the problem with Mason Greenwood, being without a player for the time being who has played regularly, and then on the other hand the club couldn't find an agreement with any other club."
Lingard has made 14 appearances this season, scoring two goals.
