The Sun says Gerrard has emerged as a shock contender for the Ipswich manager's job.
The Liverpool legend, 37, features in a list of potential successors to Mick McCarthy at Portman Road.
Owner Marcus Evans is looking at options for his next boss and is understood to have sounded out Gerrard's interest through third parties.
The ex-England captain has impressed during a year coaching at Liverpool's academy and is preparing for a future in management.
He is in charge of the Under-18s at Anfield and learning from Jurgen Klopp, while also working as a TV pundit.