Arsene Wenger says he will keep any transfer negotiations at Arsenal 'as secret as possible' in the current window.
Sky Sports understands Hertha Berlin are leading the chase to sign Lille forward Salomon Kalou.
The exciting 19-year-old is deployed heavily on the left-wing but is a versatile attacking option for Blues manager Lee Clark, who has the opportunity to extend the deal beyond the new year.
Hall, an England Under-19 international, is a product of West Ham's Academy and having made his first-team debut for the club last season has now enjoyed a total of seven appearances, six from the bench.
Clark told Birmingham's official website: "It was quite a coup getting Rob because he's been doing really well in the Development squad at West Ham which has got him on the bench for most of the season in the Barclays Premier League.
"I have got to thank Sam Allardyce for allowing this one to happen. In all honesty, I didn't think he'd allow Rob to come as he's been a big part of their squad but thankfully he has. We're now looking forward to working with him."
Meanwhile, Blues striker Jake Jervis has joined League One side Portsmouth on a one-month loan deal. The 21-year-old only returned from a stint at Tranmere earlier this week.