Congress most recently increased this limit via a binding Joint Resolution which has the force of Law if it is signed by the President.
Here is a link to the latest Joint Resolution that raised the debt ceiling. The first paragraph reads:
"Resolved by the Senate and House of Representatives of the United States of America in Congress assembled, That subsection (b) of section 3101 of title 31, United States Code, is amended by striking out the dollar limitation contained in such subsection and inserting in lieu thereof $14,294,000,000,000."Instead of revising the text to increase the numerical limit Congress could very easily just strike the whole Section 3101 from the Title 31 USC the next time the limit is near, and then there would be no self-imposed debt limit. Upon passage of the appropriation bills, the Treasury Department could just go do what was necessary to execute the appropriations.
In this regard, if Treasury continues to choose to use the Federal Reserve System as their fiscal agent, they will probably have to continue to issue US Treasury bonds due to the way the Fed runs the banking system and conducts Monetary Policy, but for no other reason such as having to "borrow the money".