Ipswich Town will reportedly demand in excess of £4 million for reported the Crystal Palace target.
Palace goalkeepers Julian Speroni and Wayne Hennessey have struggled for form this season and have each entered the final six months of their contracts.
According to the Mirror, Hodgson has put Bialkowski at the top of his list of possible replacements and has already enquired about the Pole's availability.
Bialkowski is also out of contract at the end of this season, however, it is claimed that Ipswich will still demand in excess of £4 million.
The 30-year-old initially made his name at Southampton, before moving on to Notts County and Ipswich, whose player of the year award he has won two seasons running.
Crystal Palace have also been credited with interest in Kevin Trapp of Paris Saint-Germain and Sam Johnstone of Manchester United in recent weeks.
Should Palace meet Ipswich's Bialkowski demands?