yes you can do that as it will in the same laundry room.
You only are going to need to GFI them if they're 6 feet or less from the outer edge of the sink.

A washing machine does not have to be on a GFCI. In fact, the washer might cause nuisance tripping of the GFCI. Yes, you can add receptacles in the laundry room from the existing circuit. And it might be wise to cease accepting electrical wiring advice from your previous sources.

"And it might be wise to cease accepting electrical wiring advice from your previous sources."

that made me laugh, however for some reason GFI protecting the washer receptacle seems to be something people struggle with. I have one state inspector in one area I do work in that insists that we have to GFI it. I've tried arguing and finally decided it's not worth it and always just have to remember to throw in an extra GFI on the trim.

Well, what if the washer receptacle is less than 6 ft from the sink? Is it still "exempt?" Maybe I should switch that from a duplex to a single receptacle.

When we add the laundry sink then the current extra receptacle and any others we add will need to be GFI protected because they will be within just a few feet of the sink.

Thanks for clearing up my question whether the washer needed a dedicated circuit. I thought I read that in this forum, but it was probably that the laundry room needed to be on a dedicated circuit.

Hexus IS right, under the 2008 NEC.

Any exceptions to the GFI rules for laundry rooms and basements have been removed.
If it is within 6' of a laundry sink it MUST be GFI protected. Period.
Appliances are much less likely to nuisance trip these days, and GFI's are getting smarter and less touchy. So the whole nuisance tripping thing is becoming less of an issue.

Tom, the "laundry" needs a circuit, not the washer alone. As the others said, you CAN add a receptacle to this "laundry circuit".