Jenner, 60, jetted off for a getaway on the Caribbean island with boyfriend Cory Gamble after spending Christmas with her family in Calabasas, California.
News of Kardashian, 28, falling ill broke Tuesday, with some reports citing his diagnosis as diabetes.
“Rob had stomach problems over the weekend and didn’t feel well. He was at the hospital briefly,” another source tells PEOPLE. “He has to take medicine now to control his blood sugar level. He was also told he needs to rethink his whole lifestyle. He needs to see a dietician and change the way he eats. If he doesn’t change his lifestyle, he was told that it could have serious consequences.”