The Canadian-born ex-England international is a free agent after his deal at Old Trafford expired at the end of last season and since then he has attracted plenty of interest.
"Dan [Ashworth, the Baggies technical director] has had further talks with him and the talks have been relatively positive," said Hodgson.
"We believe he's capable of playing top-flight football again and we're very interested in him.”
Asked whether Hargreaves has had a medical, Hodgson replied: "A minor one, yes. If he was to join us of course he'd have a proper medical with all the scans. We haven't done that.
"He came to visit with Alex McKechnie, the man who's been largely credited with supervising his revival and our expert here, Mark Gillett [the club doctor], looked at him and agreed.
"If the day ever came when he was even close to signing a contract, he'd have to go through a more rigorous medical, but I quite firmly believe that he would just as easily pass that.
"Alex and Owen were keen to show the doctor that the knee was fine."