Jose Junior and Matilde Junior have been pre-registered at the international school in Lugano, Switzerland, just an hour's drive from Inter's training complex at Appiano Gentile.
"Crisis at Inter: Mou to return?" was the headline in the Gazzetta dello Sport, which noted that Inter—treble winners under the Portuguese last season, including a first European Cup in 45 years—have flopped this term.
The Italian champions endured a difficult start to the season that led to the sacking of Rafael Benitez and his replacement, Leonardo, oversaw last week's Champions League exit to Schalke.
"This pre-registration in itself signifies little but there have been occasions when these signals have been a preface to transfers," wrote La Gazzetta.
"Currently, at a difficult moment, Inter fans are dreaming of Mou."
Mourinho won the league title in both his seasons at Inter and then added the Champions League and Coppa Italia last term for good measure.
He subsequently moved to Real Madrid, whom he has taken to the Champions League semi-finals as he looks to become the first man ever to lead three clubs to the trophy, having first achieved glory with Porto in 2004.
While Mourinho continues to have no shortage of admirers in Italy, he has regularly hinted he would enjoy a return to England, where he won two league titles at Chelsea but could not quite deliver in Europe.