The 37-year-old spent last season with Steve Bruce's Villans in the Championship but was let go at the end of the campaign amid their financial difficulties, which have since been resolved through new ownership.
In recent days, Villa have been linked with re-signing the Chelsea legend as they bid to return to the Premier League at the third time of asking, but according to reports in Portugal, Sporting could now hijack the Midlands side's move.
The club is currently undergoing an election for president, and candidate Pedro Madeira Rodrigues has vowed to bring in top players and a new head coach should he win the vote on Saturday.
"I can say one because I've been caught up with him," Rodrigues told the club's website of his transfer targets. "It's John Terry, a great champion, a player who does not come to win games, but comes to win games in training. I cannot pay him what he's used to, but I think he's very close to being our player."
Former Leicester City boss Claudio Ranieri is believed to be the preferred choice to take the manager's job at the Primeira Liga side.