keep watch on
keep (a) close watch on (someone or something)
To monitor someone or something closely. Keep a close watch on this pot so that the soup doesn't boil over. I've asked the manager to keep close watch on Tom. I'm pretty sure he's been stealing from the register.
Farlex Dictionary of Idioms. © 2015 Farlex, Inc, all rights reserved.
keep watch on someone or something
to monitor someone or something. Keep watch on the lady in the big coat. She may be a shoplifter. Try to keep watch on the committee's work.
McGraw-Hill Dictionary of American Idioms and Phrasal Verbs. © 2002 by The McGraw-Hill Companies, Inc.