Hodgson called an end to his four-year stint as England manager by stepping down after the nation's last-16 exit at Euro 2016 in the summer.
Before he joined the Three Lions setup, the 69-year-old had a 15-month spell as manager of West Brom, who have had Steve Clarke, Pepe Mel, Alan Irvine and now Pulis at the helm following Hodgson's departure.
According to The Mirror, the Baggies recently contacted the ex-England manager about a possible return in the event that Pulis walked away from his job over the lack of spending in the summer transfer window.
However, the club's hierarchy are now believed to be confident that Pulis, who was appointed last year, is staying put.
The report adds that Hodgson was not interested in returning to West Brom, but he is keen on getting back into management just over a month after resigning from his England post.