The Red Devils paid 34 million euros to acquire the 19-year-old left back from Southampton, agreeing to pay him a salary of €200,000-per-week, while the Blues pulled out of any prospective deal after learning of the teenager's demands.
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Mourinho instead signed Filipe Luis from Atletico Madrid in a €20 million deal and took a swipe at United on Friday, telling reporters: "To pay a 19-year-old boy what we were being asked for, for Luke Shaw, we are dead.
"We would have killed our stability with financial fair play, and killed the stability in our dressing room."
Van Gaal, however, refused to bite back in public, instead revealing that he will speak with the Chelsea boss in private.
“I don't have to answer, I don't think I have to discuss things with the media. Its better to discuss it with Jose,” he told reporters in the build-up to United's preseason clash with Roma.
Shaw made his United debut in the club's 7-0 win over LA Galaxy in midweek, featuring as a wing back in Van Gaal's 3-5-2 formation.
The Dutchman's second game in charge comes against the Serie A runner-up on Saturday and Van Gaal slammed the scheduling of the fixture, admitting he will attempt to persuade the referee to allow a drink break – as seen in the World Cup.
“It is unbelievable that we play at two o’clock,” he said. “We will adapt to the situation because all the players will play for 45 minutes except [Tom] Cleverley. For the players, it is very hard and I hope I can persuade the referee to have a drinks break as I did in Fortaleza [when Netherlands played Mexico].
“We have a sports science department and it shall be done very well with the conditions, especially those players who will play 45 minutes.”