Rep. Bob Goodlatte is the sponsor of the current GOP balanced budget amendment.
"Unless tax hikes are taken off the table, reckless lawmakers will increase taxes to pay for these new bloated spending levels, rather than bring spending in line with revenues. A 'clean' BBA provides the excuse big spenders seek to raise taxes and grow government," the groups said in the letter.
"Any lawmaker committed to restoring American solvency cannot seriously vote for a BBA that does not include a super-majority requirement for tax increases," the letter added.
Correction: Nov. 9, 2011
An earlier version of this article misstated why leadership aides want the balanced budget amendment to pass. If passed, the president would be able to ask for a debt ceiling increase of $1.5 trillion, a higher debt-cutting target.