The Daily Mail report that Hodgson would be the hot favourite to succeed Fabio Capello if the Football Association part company with the Italian after England's bad World Cup experience in South Africa.
But Liverpool are hopeful of getting the highly regarded 62-year-old Fulham manager before the end of the week.
That would leave the FA's options severely limited. It could force them to speed up a decision on Capello or approach Hodgson in a bid to make him delay a decision on his future.