The Sun says Hammers boss Slaven Bilic is keen to sign a quality central midfielder, but his interest in Carvalho has been put off by Sporting demanding £40m ($52m) for the Portugal international.
If the West Ham hierarchy fail to negotiate a deal for Carvalho, they will turn their attention to signing Wilshere on a season-long loan, according to the paper.
Wilshere has only recently returned to fitness after a broken fibula forced him to miss the final months of last season on loan at Bournemouth. He is in the final year of his Arsenal contract.
Gunners boss Arsene Wenger has invited clubs to make offers for Wilshere before deadline day, with the Frenchman saying in an interview last week that he could not guarantee a first-team spot for the 25-year-old.
"I'm open with Jack," Wenger was quoted as saying by the Guardian. "We have honest conversations. I'm open to what is the best for him.
"He is at the stage of his career where he needs to play and I can't guarantee him that today. I think I'm quite open on that."
Wilshere has made more than 100 league appearances for Arsenal since making his debut for the club at the age of 16 in 2008. However, his career has been disrupted by various injury problems and he faces leaving Arsenal on a free transfer when his contract expires next summer.
"You know how much I love Jack and his talent but today, in the football world, you need to be a consistent presence," Wenger added.
"What is at stake for Jack is not his quality, his talent or his determination – it is whether he is a consistent presence at the top. I hope he can manage to do that. Nobody questions his quality."
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