Bour rolled the ankle while racing to first base on Chase d'Arnaud's ground ball in the sixth inning.
"He's sore today," manager Don Mattingly said.
Mattingly described the location of Bour's injury as above the ankle on the outside part of his leg.
"[It was the] outside of his leg that he drug behind him, and it got twisted," Mattingly said. "He felt something on the side of his leg."
Bour has been a power threat in the middle of the order, batting .268/.347/.526 with 15 home runs and 46 RBIs.
"We don't really know how long it's going to take," Mattingly said. "We hope that when we get to New York, he's going to at least be a pinch-hitter.
"They were working on him as soon as we got here today. They're saying [they need] to get the swelling out."
Chris Johnson played first base on Sunday against the Braves at Fort Bragg Field.
Joe Frisaro has covered the Marlins for MLB.com since 2002. He writes a blog, called The Fish Pond. Follow him on Twitter @JoeFrisaro and listen to his podcast. This story was not subject to the approval of Major League Baseball or its clubs.