Keith Andrews has been told by Bolton Wanderers that he is free to look for another club.
Sky Sports understands a number of English clubs are tracking Nigerian midfielder Nosa Igiebor.
Twardzik is expected to sign as a replacement for Jason Brown, who left the SPL club on Tuesday, if a deal can be struck with his club, Italians Calcio Como.
Dons boss Craig Brown is optimistic regarding his chances of getting an agreement over the line, with it possible that no money will have to change hands.
He said: "We need cover in goal. We have got Danny Rogers, an U19 Irish international, but he is possibly not ready yet for the SPL.
"Dan is a good goalkeeper. We had him on a week's trial earlier in the season.
"If he plays another game then a payment is due to him or his previous club, so rather than have to make the payment his club are willing to let him go for free - we hope.
"He can certainly challenge for the number one spot, that's why we are bringing him in. He's a very promising young goalkeeper."