Possible blower motor failure. You can try (with the power OFF) to manually spin the squirrel cage to see if it turns.

I rent the duplex i live in and the landlord sent a maintenance guy to check it out. He just listened to it and said it sounded like the transformer. He did not think it was anything to worry about, but it sure is an annoying sound. Is he right in saying it is nothing to worry about?

Thanks for the help.

PlumbingSupply.US has a full line of Honeywell thermostats and control. They probably have the Honeywell part you need. The prices will thrill you. Nice people to deal with.

It's perfectly normal for a transformer to get louder as time goes by. If it gets too annoying then put in the request to have it changed anyway.