Lingard missed the Red Devils' FA Cup defeat to Middlesbrough after his loan move from Old Trafford collapsed on deadline day, despite Rangnick giving his blessing to leave.
The interim boss then suggested Lingard had asked for time off to clear his mind, however, the player took to Twitter to refute the claims.
But it appears everything is all patched up now with Rangnick saying there are 'no problems' between him and Lingard.
The manager confirmed the 29-year-old, as well as striker Edinson Cavani, is in contention to feature in Manchester United's Premier League clash against Burnley on Tuesday night.
On Lingard and Cavani's availability, Rangnick said on Monday: "They're both available and will both be part of the squad for tomorrow's game at Burnley. Both are top professionals, both have been training well.
"With regards to Jesse, I have a very good relationship with him. He knows I would have been willing to let him go at least until the issue with Mason [Greenwood] came up, he mentioned some personal issues.
"We have to look forward now, there are no problems whatsoever between myself and Jesse and vice versa. I'm very happy to have him in the squad and he'll be in the squad for tomorrow night's game."
Lingard is set to leave his boyhood club in the summer when his contract expires, as are Cavani and fellow midfielder Paul Pogba.
Rangnick is also set to move into his consultancy role in the summer in what could be a huge overhaul at Old Trafford. However, the German added that such a big overhaul may not be necessary.
On whether an overhaul of the team is needed, Rangnick said: "I don't know. I don't think so.
"Overall, the development of the team and performances over the last couple of weeks have improved.
"There are two points of criticism that we all have to face. One: we wasted too many clear-cut chances, 22 shots inside the box, ten were massive opportunities and we should easily have been 3-0 up at half time. It should have been 6-1 or 6-2.
"And the other one was ten seconds before we conceded the goal, it was a perfect counter-pressing situation inside their half, we took away time but not the ball away from them and eight seconds later we were outnumbered in our box. It was a handball goal and I'm still convinced this goal should have been disallowed, but we shouldn't have allowed this situation.
"Those are the two things I fault with the team. We invested a week into the game plan and many of those things have been confirmed by the team and were shown on the pitch against Middlesbrough."