Both Manchester United and Southampton have seen bids rejected by Sporting, who were also angered by his agents and took back his full economic rights from sports investment company Doyen.
The Argentine World Cup star had been suspended by Sporting as he refused to train, and he appeared to have played his last game for the club, but he has now issued a public apology via the club's own TV station.
"I was hot-headed and I'm sorry," said Rojo over his refusal to train.
"I'll be back with the rest of the [Sporting] squad tomorrow [Monday].
"Sporting signed me when I was not at my best and gave me the opportunity to play for a big club in Europe.
"I would like to thank the trust they had in me and I will resume training on Monday."
United are known to be continuing talks with Sporting, but their president Bruno Carvalho has always insisted that Rojo's buy-out clause of near £30million must be met to land him.