The Hammers have been holding out for a £3million permanent deal but Brum want to take him until the end of the season with an option to buy him then.
St Andrew's boss Alex McLeish made Davenport, 26, his No 1 target and has been chasing him throughout the transfer window.
The promotion-chasing Championship side have also run the rule over Doncaster's Matthew Mills, 22, as well as keeping an eye on Manchester City's England Under-21 star Nedum Onuoha, 22.
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