Short answer "yes"
There is a great resource for questions like this.
Michigan Coalition for Responsible Gun Owners. I'd suggest being a member but you can use the site even if not a member. They also have a weekly? email that always covers a few points of law. They almost always site the specific law that they are using. Today, I just typed "borrow" into the search field and this is what I found.
Q: A former student of mine recently contacted the Attorney General’s office to ask whether he and his wife (each has a valid CCW permit) could legally carry each other’s firearms. Could you please shed some light on this? This question has been asked and addressed on the MCRGO Forums web site but not by an attorney. Perhaps this would be a good question for the MCRGO newsletter.
A: A: It depends on if the donee has a CPL. For many years loaning handguns was not permitted. Loans of handguns had to go through the same procedures as sales of handguns. In 2001 the law changed the definition of purchaser. It dropped the "furnish" or "loan" language. At that time many attorneys took the position that loaning handguns was now allowed. There still was some concern about the "safety inspection" statute, which is Michigan's version of registration. MCL 28.429 states, "A person within the state who owns or comes into possession of a pistol shall ... present the pistol for safety inspection."
But in 2006, a new law was passed (2006 PA 75 eff. July 1, 2006; MCL §28.432) expressly stating that the purchase permit and safety inspection provisions do not apply if the donee has a CPL and the handgun is properly registered. So this new (and unnecessary) law muddies the issue enough that probably only this new law applies, not the previous position. So, after July 1, 2006, only a person with a CPL can borrow a handgun as long as it is properly registered.