The left-back has undergone an operation to remove metal bolts implanted after he suffered a broken leg back in 2015.
Rangnick had initially suggested Shaw would miss two to three weeks but has now extended the timescale of his absence.
“He will be at least another four or five weeks,” he said. “It will be difficult for him to be available for the outstanding games.”
More immediately, United face the Premier League’s bottom side Norwich on Saturday afternoon and will be without Raphael Varane and Scott McTominay, who both missed last weekend’s defeat at Everton.
Fred limped off late at Goodison Park and the Brazilian has also been ruled out, while Edinson Cavani remains out injured, too.
Better news for United is that Cristiano Ronaldo is fit despite picking up a minor knock.
Manchester United predicted XI: De Gea, Telles, Maguire, Lindelof, Wan-Bissaka, Matic, Pogba, Rashford, Fernandes, Sancho, Ronaldo