Borges, 45, was sacked on Sunday (November 1) two days after Sporting’s 2-0 league win over Casa Pia and with the Lisbon side sitting in third.
Per Fabrizio Romano, Fonseca, 53, is set to be named Borges’ replacement, leaving French club Lyon, who he joined in January 25.
Fonseca was in tears following Lyon’s 2-0 Ligue 1 win over Angers.
After the game, Fonseca told Ligue 1+: "Tomorrow I will be here, with the team. For now, there is nothing.
"Leaving? It's a possibility,” he admitted. “I really like this group; I think they are wonderful. Sometimes it's difficult to understand the difficulties we are facing.
"What this team has achieved so far is incredible. No one could have imagined that, with all these difficulties, the injuries and the departures, we would be where we are.
“It's not an easy time, but we are here doing the work. We are a family.
"That's why we are getting these results. People need to understand that things are not easy for everyone at the club."
"I think the president and Matthieu Louis-Jean, the sporting director, have done a great job, but there also needs to be some understanding, because this is not an easy time," he continued.
"At the moment, anything is possible," he replied when asked whether he could say no to his beloved Sporting.
"I don't know. I really like being here - the club and the staff. It's difficult to find people like the ones we have at the moment. I don't know if I will be here in the coming days. Leaving during the winter break? Tomorrow I will be here with the team."