Hodgson has revived Albion's fortunes during his 15 months in charge and they are guaranteed a third successive year of Barclays Premier League football for the first time.
Robson said: "I think Albion would want to keep Roy. You need a consistent basis at the club and Roy has established that basis since he has gone in."
He added: "They did well to climb out of the relegation area when he went in, steadied themselves in the league, and this year has been a good solid season for them. There is a good base there, and if you lose the manager, all of a sudden that might be difficult to replace."
But Robson, a member of the successful Albion side during the Ron Atkinson era, is unsurprised Hodgson is the subject of speculation linking him with England.
He said: "Roy is an experienced manager, has the experience of being an international manager with Switzerland which holds you in good stead. The FA aren't giving too much away as to whether it is going to be Harry Redknapp or is it going to be Stuart Pearce?
"I know that Roy seems as if he would be happy to go into that role if he was asked. I'd think he'd do a good job. Most managers could do a reasonable job as you are working with the best players in England.
"Fortunately for me, when I worked with Terry Venables for two years, it was a joy working with people like Shearer, Sheringham, Adams, Campbell, because you are working with the best players.
"I could see why Roy wouldn't turn it down if he was offered the job."