Her stance is in line with President Obama, who said earlier this week that he is not aiming to "chase a balanced budget just for the sake of balance" over the next decade.
"My goal is how do we grow the economy, put people back to work, and if we do that we are going to be bringing in more revenue," he told ABC News on Tuesday.
The budget plan produced by House Budget Committee Chairman Paul Ryan (R-Wis.) does achieve balance in 10 years. But while his plan repeals most of the healthcare law, it takes into account savings from the program.
Overall, the Sessions amendment would slow the growth of spending to an average of 3.4 percent a year, cut taxes by $923 billion, lower non-interest spending by $3.86 trillion, reduce interest costs by $546 billion and lower the debt by $3.5 trillion.
He said that his amendment, which is expected to get a vote on Thursday afternoon, permanently boosts economic growth by reducing gross debt below the 90 percent "danger zone" by 2020.
"By the time we balance the budget, the gross debt would stand at 80 percent of GDP," the amendment says.
Johnson's amendment would establish a point of order against a budget resolution that does not achieve a budget surplus after 2022.