Is it possible to make an offest a construction line?

I have wanted that before too. I don't think you can do that, but what I have done is offset and make the base construction, then right click on a segment from the offset and select chain. Then you can make the entire offset construction at once. It isn't that much trouble.

As Jamil said, the answer is NO. The other workaround might be to convert the geometry into construction line and then do the offset.

I would recommend you you to please raise this requirement as Enhancement Request (log into your SOLIDWORKS customer portal and click on Enhancement Request) stating why this would be helpful.