Matthews still has a year remaining on his Bolton contract but, with the Trotters currently in administration and set to start next season with a 12-point deduction in League One, he could potentially move on.
The 25-year-old, who began his career with Norwich City but never made a senior appearance for the Canaries, is understood to be targeted in addition to Gillingham's Tomas Holy who signed on Friday.
It is likely to be a summer of change between the sticks at Portman Road, with a number of Championship clubs understood to be interested in Bialkowski and Dean Gerken released at the end of his deal.
Birmingham have been linked with a potential move in the national media, having attempted to sign Bialkowski for £3million last summer.
The St Andrew's side are understood to remain admirers of the 31-year-old but, given Garry Monk is likely to be limited to free transfers and loans this summer, he's likely to be beyond their financial reach.
A handful of clubs are understood to have made enquiries regarding Bialkowski but are yet to make formal offers.
Should Bialkowski move on, with the Pole open to the idea of a fresh challenge after five years in Suffolk and a number of Championship clubs understood to be interested, Matthews could be a prime target.
The Blues are understood to have compiled a long list of goalkeeping targets this summer with Exeter's Christy Pym, who has since joined Peterborough, understood to have been looked at by the Blues.
Town boss Paul Lambert is understood to be keen to carry three goalkeepers in his squad next season, with Holy and academy graduate Harry Wright currently on the books in addition to Bialkowski.