Sørloth signed for the south London team on transfer deadline day for a reported initial £8.8 million (10 million euros). But that fee could rise to as much as £16.5 million (18.8 million euros, 141 million Danish kroner), according to Danish newspaper BT, which described the transfer as a 'crazy deal'.
"FC Midtjylland are being well paid for one of the Superligaen's most dangerous strikers. Absurdly well paid," the newspaper wrote on Wednesday evening.
Another remarkable aspect of the transfer is the fact that 22-year-old Sørloth, who has 12 caps and one goal for Norway, has only been at Midtjylland for half a season.
The Jutland club signed him for £300,000 last summer, with his previous notable high-level experience coming in the Dutch Eredivisie, where he scored just six goals over two seasons for FC Groningen.
But a total of 15 goals in 26 European and domestic games for Midtjylland this season was good enough to get the striker a move to the Premier League.
"Crystal Palace were willing to pay an exceptionally high amount to bring Alexander to [their] club, and we cannot reject a sale of that magnitude. As a business, we must show leadership and act responsibly in such a situation, and the sale gives an economic foundation and stability on which FC Midtjylland can develop," the Danish club's CEO Claus Steinlein told BT.
"It's a dream come true to play in the Premier League," Sørloth said following the transfer, according to the BBC.
"I've dreamt about this since I was six," he added.
The striker will wear the number nine shirt at Palace.