Context is everything
For a Background Verification, the important thing is who signs your paycheck and legally employed you, and who can verify this information. For the Employer, in this frame of reference, is ABC. The contact information should be the official contact information of the company. Their website is a fine place to get this from. Make sure to give the number to the front desk, rather than your supervisor's direct number, even if you have it. The front desk would be able to route your call to the appropriate HR rep or your supervisor per their policies, or should your supervisor be out.
Supervisor is a bit trickier to figure out what the context is. However, for a background verification, you typically want the person who can verify your employment at the contracting agency, and speak as a representative of that company. For example: I have been on a contract in the past where I was hired by Contracting company A, working for company B, on a contract shared with company C. I had a 'manager' at company A, a separate executive managing the contract, a manager at company B who owned the contract, and a direct supervisor/floor manager who oversaw my day to day tasking (who was an employee of company C). In such a situation, depending on the context of what this background was trying to accomplish, the correct answers could be my direct manager (in context of my employment with company A), the contract manager (in context of the contract itself), or the direct supervisor (in context of my job duties and skillset). If I had to outline this position on a background verification, I would put either my direct manager, or the contract manager, as both are representatives of the company that hired me, and both could speak to my position on the contract and at the company.
For a Resume, the important thing is who you actually performed work for, as you'll be talking about your job duties and skills performed on the job. The supervisor should be the person who you report to and reviews your work, as they are the best capable to speak on behalf of those skills.
You do not want to misrepresent yourself as a member of the company though. If you have only a single contract with the contracting agency, you can put the company you worked for, and then put the reference as contractor:
Position at XYZ (contracting through ABC) 10/2016 - 12/2017
While if you have multiple projects through a contractor, you could also put the contracting company first, then list the contracts underneath that.
- Contractor at ABC 10/2016 - 12/2017
- Position - XYZ 05/2017 - 12/2017
- Position - UVW 10/2016 - 05/2017