The defensive midfielder has been without a club since being released by Manchester United at the end of last season.
Hargreaves, 30, made just 27 Premier League appearances in four seasons at Old Trafford due to a persistent knee injury.
Should the former Bayern Munich player be able to prove his fitness then Hodgson is prepared to offer a contract, although the boss expects competition.
"Obviously, we're monitoring a player of his quality," Hodgson told Sky Sports News.
"But if a situation develops whereby it's clear he is no longer injured and is still capable of playing Premier League football, then I don't think we'd be the only team interested in his services."
Hargreaves joined United in 2007 for a fee of £17m.