The injury-prone ex-England international was released by United at the end of last season and is still on the hunt for a new club.
Hodgson has already shown interest in Hargreaves this summer and is again curious but admits it is a tough call as it is unknown if the 30-year old is capable of playing at the top level due to his ongoing injury problems.
"Obviously, we're monitoring a player of his quality," the Baggies boss told Sky Sports News.
"But if a situation develops whereby it's clear he is no longer injured and he is still capable of playing Premier League football, then I don't think we'd be the only team interested in his services.
"You'd have to check with (technical director) Dan (Ashworth) if there's to be any future communication because the big question with Hargreaves is not can he play football, but is he going to be fit again.
"It's not possible for my naked eye to make a decision on that."