In the closing stages of Atletico's 2-0 victory over local rivals Getafe, Costa's left knee struck a goalpost as he scored his team's second goal and the 25-year-old was stretchered off with a large cut amid fears he may have broken his leg.
Atletico later confirmed that Costa had not broken anything, but immediately after the game Godin revealed his hope the key striker could remain fit for the rest of the season.
Costa missed Atletico's UEFA Champions League semi-final second leg last week with a hamstring strain after limping off in the first leg.
"I don't know (exactly) how bad it is but he has a cut around the tibia. Hopefully that's all," Godin said.
The win over Getafe saw Atletico to 82 points, three clear of local rivals Real Madrid at the top of La Liga, while Barcelona are third on 78.
Godin and Costa scored the goals against Getafe and, afterwards, the Uruguayan central defender emphasised the importance of the result.
"It was important to retain a big difference between Madrid and Barcelona," Godin said.
"A win is a win but there's a bitter taste because of Diego's injury."
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