Lingard later posted on social media to say it was the club who had advised him to take a break.
The 29-year-old missed the FA Cup defeat to Middlesbrough on Friday but Rangnick said he will be available for the trip to Turf Moor while insisting the pair remain on good terms.
“With regard to Jesse, I have a very good relationship with him,” Rangnick told a news conference on Monday.
“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.”
Rangnick will also have Edinson Cavani available after being granted extra time off at home in Uruguay.
Raphael Varane should also be fit to face Burnley despite coming off early against Middlesbrough.
“He seems to be physically fit, has no issues anymore with injuries,” Rangnick said.
“Up until he was substituted [against Middlesbrough] he did extremely well with Harry [Maguire]. I’m very happy to have him for tomorrow’s game, hopefully, we can have him in the same form for the upcoming games.”
United head to Burnley in fourth place in the Premier League table and a game in hand over fifth-place West Ham.
Despite the setback against Middlesbrough, United have won their last two league games and a third in a row at Burnley would move them to within six points of Chelsea in third with a game in hand.
“Overall, the development of the team and performances the last couple of weeks have improved,” Rangnick said.
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