The nearly $1.4 trillion student-loan market is expected to be far more hospitable to private lenders under a Donald Trump administration than during the last eight years.

Under President Barack Obama, banks such as J.P. Morgan Chase & Co. and U.S. Bancorp—student-loan lenders until a few years ago—were largely sidelined. Origination of federally backed student loans was transferred entirely to the U.S. government, already the dominant force in the market.

The...