Shaw became the fourth-most expensive defender in history this summer when he signed for Manchester United for an initial fee of £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 £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 quoted 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.
After signing Diego Costa, Filipe Luis, Cesc Fabregas and Didier Drogba, Mourinho is happy with his squad ahead of the new season.
That is not the case for Van Gaal, who admitted he has been unimpressed with some of United’s players in training and has been laying down the law on the eve of today’s friendly against Roma in Denver.
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.