The Black Cats are believed to be willing to let Fulop leave the Stadium of Light following the summer arrival of Simon Mignolet.
Fulop ended last season on loan at Manchester City and, despite the injury to Craig Gordon, it appears he does not figure in Steve Bruce's plans.
Ipswich boss Roy Keane is keen to strengthen his goalkeeping options after allowing Richard Wright to leave Portman Road.
Keane is a big admirer of Fulop having worked with him during his time in charge of Sunderland and it is believed he has made a bid of over £1million for the Hungarian shot-stopper.
Fulop is thought to be considering the move, but is weighing up his options to see if any other interested sides come to the table.