Porto confirmed Casillas, 37, was taken to hospital but is now in a stable condition, his "heart problem resolved".
Azpilicueta represented his country alongside Casillas, who is Spain's record appearance maker with 167 caps, and offered well-wishes at a pre-match news conference ahead of Chelsea's Europa League semi-final encounter at Eintracht Frankfurt.
"It was the first thing I wanted to say before any question and, obviously, I want to wish him a good recovery," he said.
"The news has shocked us, it is quite tough. But we have been told that he is doing well, he is improving.
"He has been my team-mate in the national team for quite a few years.
"From everyone in Chelsea we want to wish him a good recovery and we hope to see him very soon on the pitch."
Casillas won five La Liga titles and three Champions Leagues during 16 decorated seasons with Real Madrid, while he was Spain's last line of defence during their golden period – winning Euro 2008 and Euro 2012 either side of the 2010 World Cup.
The goalkeeper also offered an update on the situation from his hospital room, saying all was well.
"Everything under control here, a big scare but my forces are intact. Thank you very much for the messages and affection," Casillas said via social media.
Casillas has been ruled out for the rest of the season, with Porto president Jorge Nuno Pinto da Costa having also made it clear that the issue would require rest and recovery before he would be able to return to the pitch.
Porto have just four matches left in the season, with three league clashes and a Portuguese Cup final against Sporting still to come.
The club sit just two points back of league leaders Benfica as they head into the closing run.