Last season's defeat play-off finalists have added four fresh faces to their squad so far this summer, with Orjan Nyland and John McGinn joining permanently and Andre Moreira and Axel Tuanzebe completing season-long loan moves.
The loan window remains open for Championship clubs until the end of the month and manager Steve Bruce remains keen on further additions.
According to the 'Birmingham Mail', Abraham remains a priority target and that there have been encouragin signs in recent days to suggest a deal could be struck.
Bruce himself gave an update on his pursuit of fresh blood, saying: "The big clubs want their players to play. Chelsea have got 31 players out on loan, for example, it's incredible. Our aim is to make a breakthrough or two in the next week.
"The owners have preached from the start that we want a younger outlook so we can grow and get better. That always takes time.
"I think it's fair to say, that because of the financial restrictions we had last year, we were ageing. We had to get frees and loan players.
"You aren't going to get many 23 or 24-year-old's in that bracket. That ageing squad nearly got us there (to promotion). But the way forward, certainly, is to be more energetic - and we want the players to be ours.
"The remit going forward is to be younger and fresher. We need them to be permanent signings, though, really rather than putting the stick and plaster over it like we have done in the past 12 months or so."