Nastasic, who plays for Bundesliga side Schalke, did not specify when or how he got injured. He was substituted in the latter stages of their 3-2 defeat at Hamburg SV on Saturday.
His absence would be a huge blow to Serbia’s hopes of reaching the knockout stages of the 32-nation tournament in Russia, with coach Mladen Krstajic devoid of enough quality options at the back.
Serbia, who qualified for their first major tournament since the 2010 World Cup, are in Group E alongside Brazil, Switzerland and Costa Rica.
They start the campaign against Costa Rica on June 17 in Samara, followed by games against the Swiss in Kaliningrad on June 22 and Brazil in Moscow five days later.