Our bank wants to create a new money market and savings account but wants to attract new money, not current customers moving money they currently have here to the new account. So they want to have the restriction of the new account for only money received from a check drawn on another banking institution. Can we do that?

I have seen other banks do it all the time.

We did that often at a prior bank I worked with. It was disclosed in all print and radio advertisements.

Yes, you can do it - however be prepared to transfer your 'good' customers into this product. I've worked at institutions who promoted accounts for 'new' money only, but most of the funds in these accounts was transferred from existing customers.