The Posh's director of football, Barry Fry, has confirmed that the relegated Hammers have asked about the Scotland international but have not spoken about teammate Joe Lewis.
"West Ham have not spoken to me about Joe Lewis," said Fry.
"But they did ask me what we wanted for Craig. They have not made a bid as yet, but they are having a serious go at getting back into the Premier League straight away.
"One Premier League club who have spoken to Craig set him a deadline of today to make his mind up, but he will not be going there. It's not a question of money, it's a couple of other things that he's not happy about.
"Craig is right not to rush into anything because there is still plenty of interest in him."