Almost all of the Postdoctoral positions requires to attach publications while contacting. Can a manuscript under review be sent to address this?
You can send it if you clearly label it as under review and not yet accepted. Name the journal reviewing it as well. 

I don't see any problem with this as long as you are honest about its current status. Of course you should also include other publications if available. But in your labeling of this paper you can give your personal assessment of its importance in your work. People can do their own evaluation.