Finding out whether a title company will insure a property

I recently purchased a property at auction.? Before fixing up the property I would like to find out whether a title company would be willing to insure it for a new buyer.? Is there a way to get a title company to do the necessary research?? Would they charge for this or just do it based on the opportunity to insure the company when it is actually sold?? Is another option to tell the title company we would like to insure the property now.

You can open escrow before the property is in contract. As a standard part of opening escrow the title company will do a preliminary title report.
As for having them insure you, you will need to talk to them. It can be problematic. They may be willing to do it, if you are willing to except title exceptions around issues arising from the auction itself (for example they would not want to insure you against a lawsuit from th former owner).