McLeish said: "Lee's a dynamic player and I want to bring a dynamic look back to the team.
"Roy Aitken, one of my assistants, has worked with him at Leeds and knows him as an attacking midfielder but he also knows that Lee can give a lot to the team as well."
Bowyer has struggled to hold down a first-team place at Upton Park this season and has only featured in seven games for Gianfranco Zola's side.
The former Leeds and Charlton midfielder joins Matthew Etherington on his way out of the club. Etherington completed a £3 million move to Stoke on Thursday, while Calum Davenport is on the verge of joining Bolton.
West Ham, meanwhile are continuing to hold on to Craig Bellamy and Scott Parker and have already rejected a join bid from Manchester City.
Luis Boa Morte has been linked with a £1.5 million switch to Hull City, but Phil Brown has denied reports that a deal has already been done.
"Speculation will follow us throughout this month and Luis is just another one in the line of players we're supposedly interested in.
"But as far as we're concerned, there's nothing in it."
Following Etherington's move to Stoke, Boa Morte is currently Zola's only natural option on the left wing.