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The 21-year-old suffered the injury during the first half at the City Ground but was able to make it out for the second half with his right thigh bandaged.
However, Palmer was forced to pull up just six minutes into the second half after suffering further pain to his right thigh.
Palmer was able to walk off the pitch unaided and was seen removing the bandage as he made his way towards the touchline.
The forward is now a doubt for England Under-21s’ European Championship qualifier against Ukraine on Monday.
Speaking after his side’s win, England Under-21s head coach Lee Carsley said: ‘There are a few bumps and bruises and we will monitor Cole but we will not be taking any chances.’
Palmer joined Chelsea in a £42.5 million deal from Manchester City in the summer transfer window.
He has made six appearances for Mauricio Pochettino’s side, scoring one goal and registering two assists.
Meanwhile, England Under-21s recorded their biggest win since their 9-0 win over San Marino in 2013 with their 9-1 victory over Serbia.
Jaden Philogene, Harvey Elliot and Noni Madueke all scored twice. Liam Delay and Jonathan Rowe also found the net while Luka Subotic scored a late own goal.
The win keeps England top of Group F with three wins from three games.
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