"I can't say who he is at this stage, but we're hopeful of getting him," Lawrence told the Western Mail.
"If it happens, though, it won't be until the end of the week because the issue is complicated."
The player is thought to be a Premiership midfielder, a similar situation to the one that saw Gary O'Neil come to Ninian Park.
City have already strengthened with the addition of Neil Harris from Millwall. The striker came onto the pitch for the last 15 mins v Gillingham replacing the impressive Cameron Jerome.
Graham Kavanagh has now served his three match ban so is available for selection for the game against Sunderland at Ninian Park this coming Saturday.