Costa has been a revelation since moving to Stamford Bridge from Atletico Madrid in July, scoring nine goals in seven top-flight matches despite being troubled by a hamstring problem.
The 26-year-old scored his first international goal for Spain in a Euro 2016 qualifier against Luxembourg last weekend, but could miss Chelsea's clash with London rivals Palace.
Mourinho also has concerns over Ramires [muscle problem], Didier Drogba [ankle], and Andre Schurrle [illness], while Jon Obi Mikel is a doubt for a game which comes only three days after he featured for Nigeria against Sudan.
The Chelsea manager said at a press conference on Friday: "We have some problems, yes. Ramires and Drogba, Diego, Schurrle, Mikel."
When asked if Costa, the Premier League's leading scorer, is a big doubt to face Neil Warnock's side, Mourinho replied: "Yes."
He added: "The Spanish team played in Luxembourg on Sunday. I gave them Diego, [Cesc] Fabregas and Azpi [Cesar Azpilicueta] two days off [Monday and Tuesday], so they had to report Wednesday at 3pm.
"Diego was here Monday at 9am. That shows you how strong he is."
Mourinho refused to comment on whether Costa had reported to the club on Monday to receive treatment.
The former Real Madrid coach also did not apportion blame to Spain boss Vicente del Bosque after Costa played the full 90 minutes in a defeat against Slovakia and was substituted just eight minutes from time against Luxembourg three days later.
"They [national team coaches] have the power. We don't have the power." he said.
"I don't speak with national team managers unless they want to speak with me. I'm powerless because I'm not there. I don't waste my time contacting national teams."
Thank you for reading 5 articles this month* Join now for unlimited access
Enjoy your first month for just £1 / $1 / €1
*Read 5 free articles per month without a subscription
Join now for unlimited access
Try first month for just £1 / $1 / €1