Rangnick suffered his first defeat at United in a 1-0 defeat to Wolverhampton Wanderers on Monday, and defender Luke Shaw said after the match that his side had not felt “together” on the pitch and questioned his teammates’ intensity.
Sources have told ESPN that Rangnick was disappointed with the Shaw’s comments but not surprised, with the manager concerned over dressing room unrest.
“We’ve all been disappointed with recent performances and we’ve been as disappointed with the news coverage questioning our commitment to not only the manager and the coaching staff but the club,” Rashford said in a statement on Twitter.
“I have endless respect for both and I’m looking forward to bettering my game under the staff… I’m not sulking, I’m not unhappy.
“Am I disappointed in some recent performances? Of course I am. I’m my own biggest critic. It’s been a tough start but I am determined to prove my worth. My dedication and my desire should never have been in question. I love this club.”
The England international came off the bench in Monday’s loss to Wolves that left United in seventh place, four points off the Champions League places but with a game in hand.
Rashford, 24, missed the opening months of the Premier League season with a shoulder injury but has since made 14 appearances in all competitions, scoring three goals.
Rangnick told a news conference on Monday there are “unhappy” players at United due to a lack of playing time, and sources have told ESPN a number of fringe players want to leave the club.
“When you have a big squad, in the last two games, we had most players available and only 10 field players can play and three substitutes,” Rangnick said.
“We have players with contracts expiring in the summer, we have maybe also one or two players who want to leave and are under contract.
“It is about the players dealing with that in a professional way, to show in training, to show up and get a chance to play.
“If this is not the case, the player, club and agents need to discuss the situation.”
United host Aston Villa in the third round of the FA Cup on Monday.
Information from Reuters contributed to this report.