Shaw became the fourth-most expensive defender in history this summer when he signed for Manchester United for an initial fee of B£27million.
Chelsea, who Shaw supported as a youngster, were interested in the 19-year-old, but Mourinho revealed he pulled out of a bid to sign the player due to his wage demands.
The Chelsea manager claimed Shaw's reported wages of around B£100,000 a week were way too large for a player who only made his Premier League debut in November 2012.
"If we pay to a 19-year-old boy what we were being asked for, to sign Luke Shaw, we are dead," Mourinho was quotes as saying in several national newspapers.
"We would have killed our stability with financial fair play and killed the stability in our dressing room, because when you pay that much to a 19-year-old kid - a good player, fantastic player - but when you pay that amount of money, the next day, we would have had players knocking on our door.
"They would have been saying, 'How is it possible I play 200 games for this club, won this and that, yet a 19-year-old comes here and gets more money than I get?'"
Van Gaal refused to get drawn into a public war of words with Mourinho - a former protege of his at Barcelona - but it appears he will discuss the matter privately with the Portuguese.
"I don't think I have to discuss things in the media, it's better to discuss it with Jose," he said.
Mourinho recently spoke about how he was looking forward to coming up against the man he assisted at Barcelona for three years.
The Chelsea boss said earlier this week that there is a "fantastic relation" between the pair, but that could be tested further this season if the Portuguese criticises Van Gaal's new charges.
That is not the case for Van Gaal though. The Dutchman is looking to sign at least one centre-back before the transfer window shuts and another midfielder.
One of those on Van Gaal's shortlist is Kevin Strootman. Roma coach Rudi Garcia insisted on Friday that the Holland midfielder would not leave the Italian club this summer.
But Van Gaal refused to rule out the possibility of signing the 24-year-old once he recovers from the serious knee injury he suffered in March.
"We have to wait and see how he comes back from his injury, he has been out for over six months," Van Gaal said when asked about reports he was interested in signing his compatriot.
Strootman has been left in Boston, where Roma played Liverpool earlier this week, to recover from his injury, so the Dutchman will not face United in Saturday's friendly in Denver.
Van Gaal, who admitted he has been unimpressed with some of United's players in training, was laying down the law to his squad on the eve of the match.
The Dutchman gave Chris Smalling a dressing down during an open training session at the Sports Authority Field and he seemed extremely unimpressed by a select group who continued to miss the target from set-pieces.
Off the pitch Van Gaal has been making his presence felt too, demanding water breaks during Saturday's match, which is due to be played in the 33 degree Colorado heat.
"It is unbelievable that we play at 2pm," said Van Gaal, who criticised FIFA for making Holland play Mexico on a blistering afternoon in Fortaleza during the World Cup.
"We have to adapt to the situation.
"All the players will play 45 minutes apart from Tom Cleverley and Tyler Blackett.
"For these players it is very hard.
"I hope to have a pause for a drink also. I shall convince the referee, like it did in Fortaleza. It was the first time they had a drinks break at the World Cup. Mostly they listen."
Cleverley lost his place in England's World Cup squad after a poor season under David Moyes, but hopes to excel this term.
The midfielder, who got two hefty pats on the back from Van Gaal as he sat by him during the pre-match press conference, thinks he can benefit from playing in the manager's new 3-5-2 formation.
"I watched Holland in the World Cup and I think I can be his type of player," Cleverley said.
"I have to show that in training and most importantly in the games. I think there is a place for me. I have to prove that."
Rafael will miss the friendly, which is part of the International Champions Cup, after he hobbled off in training with what looked like a groin injury.
- Sports & Recreation
- Van Gaal
- Manchester United