The Danish rider came down heavily on Saturday, suffering abrasions to his forehead and face as well as the broken bone.
Cort also broke his collarbone in a crash at this year's Tour de Yorkshire, though it has not yet been revealed if it's the same one.
The 24-year-old has signed for Astana from Orica-Scott on a two-year deal, though he will now be unable to take part in the team's first winter training camp in a week's time.
"Unfortunately, Astana 2018 newcomer Magnus Cort Nielsen crashed during a training and broke his collarbone," read an update from the team. "We wish to him a very fast recovery. Get well soon, Magnus, we are waiting for you!"
A fast finisher, Cort won two stages of the 2016 Vuelta a España, including the final leg in Madrid, and also rode regularly in the service of Caleb Ewan at Orica-Scott. The 24-year-old was also part of Orica-Scott's Classics squad in each of his three years at the team, and he will look to progress further on the cobbles at Astana.
"Joining Astana next season is an exciting challenge for me and I hope I will have a chance to show my ability in the big Classic races, and, of course, also help the team in the Grand Tours," he said when the deal was finalised.