The 32-year-old left-back sustained the problem during his side's 3-1 away win against Norwich City in the Premier League and was replaced in the 75th minute.
The injury is likely to raise concerns among the England supporters whether Cole will be fit to play in the Three Lions' upcoming World Cup qualifiers against Montenegro and Poland, but Mourinho is hopeful that it is not serious.
"When I took Cole off, I didn't know [he was injured]," he told reporters. "He has something in the ribs that they need to assess but hopefully it is fine. I hope he goes with England.
"I think he fell in a difficult position. He now goes for a scan on his ribs and I hope he is fine."
"He's my concern because I want England to do well and my players to do well in the national team," added the Portuguese. "I will be there on Friday [to watch England]."
England, who are currently at the top of Group H with 16 points from eight matches, will host Montenegro at Wembley on Friday.