According to the Real Estate Settlement Procedures Act, it is illegal for anyone involved in a mortgage transaction to require the borrower to use a particular service provider in connection with the loan. So, the short answer to your question is “yes.” A borrower can determine which title company to use for a residential refinance transaction. That being said, the short answer requires some qualification.
In the title industry, like every other industry, there are some bad apples. A particular title company or closing agent might not record documents promptly, they might do sloppy work — some have even been known to participate in fraud. In the interest of sound business practices, lenders keep track of this and maintain lists of who’s been “naughty” and who’s been “nice.” So, while they can’t require you to use Acme Title Company, there is nothing which says that they can’t refuse to allow the utilization of Generic Title Company’s services if they have had bad experiences with them in the past.