The 30-year-old was released from Old Trafford over the summer following an injury-plagued spell with the Premier League champions, where he made 39 appearances in four years.
The Canadian-born midfielder recently passed a medical with the Midlands club.
"Whether or not our offer is going to be interesting to him - that's what I don't know," said Baggies manager Roy Hodgson.
"I know Dan [Ashworth, West Brom technical director] has emailed a proposal to Owen but I don't know what the proposal is.
"We made it clear, 'Look we've done the test, we agree with you that the knee's okay and you can play football. And of course if you'd like to come to play for us that would be lovely'," he added.
Hargreaves joined Manchester United from Bayern Munich in 2007 for £17m.