After handing latest signing Sead Haksabanovic a late but impressive debut in Wednesday's 4-1 Premier Sports Cup victory over Ross County, Postecoglou refused to rule out a move for the attacker's former Rubin Kazan team-mate Abildgaard.
When quizzed about reports linking Celtic with the powerful Denmark midfielder, Postecoglou said: "We have been linked with a lot of names. There's 24 hours to go and there are some outgoings we are working on as well.
"Fans can expect some transfers and ins and outs."
Albion Ajeti joined Sturm Graz on a season-long loan with a view to a permanent move and Mikey Johnston is set to depart on a temporary basis.
"We are still working on Mikey to get him somewhere and there's a couple of good possibilities for him," Postecoglou said.
Rangers boss Giovanni van Bronckhorst declared himself "relaxed" about the impending deadline on Tuesday.
Niko Katic left Ibrox for Zurich on Wednesday having failed to get into the manager's plans, while the future of the likes of Kemar Roofe could also be uncertain.
Hearts are set to act following injuries to Liam Boyce and Kye Rowles.
After cup defeat by Kilmarnock, manager Robbie Neilson - who ruled out a move for Louis Moult - said: "We knew we'd to get bodies in anyway. I don't want to put a number on it but you can see that we certainly need a few to give us more because after this we go Saturday-Thursday-Sunday-Thursday for about six weeks."
Kilmarnock manager Derek McInnes is happy with his squad after signing striker Christian Doidge on loan from Hibernian.
"That might be us," McInnes said. "The board have been really supportive. It's not easy doing as much as we had to do.
"You can never say never. Whether one or two of ours go out or whatever or maybe one might come up late on, I'd still maybe like to do one more but if the window shut with what we've got done, getting Doidge in makes me think I can sleep a bit easier at night."
Hibernian were also linked with a move for Burton Albion striker Moult following Doidge's exit.
Moult's former club Motherwell are still on the hunt for players while Aberdeen could do more incoming business, especially if Christian Ramirez and David Bates move on, although manager Jim Goodwin stressed there was no panic.
St Johnstone boss Callum Davidson was looking to sign two players this week and got one on Wednesday with the purchase of Nicky Clark from Dundee United.
United declared themselves finished last week, while Livingston and St Mirren are also not expected to bring anyone in.
Ross County are also unlikely to add to their 10 summer signings.
Manager Malky Mackay said: "We did our business early. If I was bringing somebody in he would have to come into the starting team and they're not easy to find."