Democrats, some Republicans and Treasury Secretary Timothy Geithner fear that if Congress does not raise the the debt ceiling the U.S. could default on its financial obligations to debtors, including China, which could prompt a global crisis.

Cornyn, the chairman of the National Republican Senatorial Committee, has been a high-profile proponent of a balanced-budget constitutional amendment that would force the U.S. to spend only the amount it receives in revenues, except in times of war or when three-fifths of the House and Senate vote to spend more.

In March the Senate fell short by two votes of passing a non-binding "sense of the Senate" resolution calling on Congress to pass that amendment.

Some Republicans are hoping to use the possibility of a Democratic balk on the debt-ceiling fight to force spending reductions, and possibly a constitutional amendment, as a remedy to the government's deficit problems.

Cornyn was speaking at groundbreaking event Monday in Cibolo, Texas, as reported by KSAT, a local ABC affiliate.