Hargreaves has been a free agent since being released by Manchester United at the end of last season following four-injury laden years at Old Trafford.
The Baggies were the first club show an interest in the former England midfielder soon after his release and have been in contact with the 30-year ever since.
When quizzed on whether the midfielder could withstand a more stringent medical Hodgson was positive he would pass with flying colours.
He said: “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.”
“We believe he’s capable of playing top-flight football again and we’re very interested in him.”
Despite aiding him in his rehabilitation Hodgson expects Albion to face competition from other Premier League clubs for a man who cost Sir Alex Ferguson £17 million in 2007.