Goal exclusively reported in August that Arsene Wenger’s side submitted two separate bids during the transfer window for the Portugal international, who has also linked with Manchester United.
The 22-year-old admits he is content to remain with Sporting for the short term but insists “sooner or later” a switch to either the Premier League or La Liga will happen.
"I would not say I was disappointed [to stay], but despite being very happy at Sporting, we all have ambitions and I hope one day to be playing in the best leagues in the world, the English or Spanish,” Carvalho said in an interview with the official magazine of the Portuguese Football Federation.
“It did not happen this year, but if I continue to do my job as I have been doing for sure that sooner or later it will happen.
"I understand the proposals, but I think that there was no value for the clause and therefore they have been refused."