Both clubs confirmed last week that an initial offer for Ridgewell, who can play at left-back and centre-half, had been rejected.
But sources close to the Baggies - as well as Blues - are adamant a fresh, improved offer has not been lodged.
Albion insist the club's stance has not altered from head coach Roy Hodgson's media conference on Friday when it was indicated they would "move on" from attempting to sign Ridgewell.
The 27-year-old stressed his desire to leave St Andrew's last summer amid interest from Barclays Premier League clubs. City, however, turned down a transfer request from the versatile defender which came within 24 hours of the close of the window.
Meanwhile, Birmingham winger Jean Beausejour is yet to complete his move to Wigan. The Latics agreed an undisclosed fee for the Chile international on Monday and he is now discussing personal terms at the DW Stadium, as is Bristol City striker Nicky Maynard.