Elon Musk’s comments about taking Tesla Inc. private are threatening to increase the electric-car maker’s borrowing costs, presenting a fresh challenge for a company already dealing with a dwindling cash pile.

Tesla’s only unsecured bonds, a $1.8 billion note issued last August with a 5.3% coupon, have bounced around since Mr. Musk first tweeted his interest in a go-private transaction last week. The most actively traded version of those bonds changed hands late Monday afternoon at 90.750 cents on the dollar for a 6.994% yield,...