That would take the place of the 6.2 percent the workers now contributed to Social Security.

Another 6.2% is sent to Social Security by employers. Under the Sessions bill, employers would continue to make this matching contribution to Social Security, but after 15 years, employers could also send that amount to the employee's SAFE account.

Sessions said this transition to a private retirement savings option is needed because Social Security last year began paying out more money than it took in.

"Our nation's Social Security Trust Fund is depleting at an alarming rate, and failure to implement immediate reforms endangers the ability of Americans to plan for their retirement with the options and certainty they deserve," Sessions said. "To simply maintain the status quo would weaken American competitiveness by adding more unsustainable debt and insolvent entitlements to our economy when we can least afford it."

Under the bill, employees would be able to make tax free contributions to their SAFE account, and take tax-free distributions at retirement age. The bill would also allow employees to stay with the Social Security program if they wish.

Other sponsors of the bill are Reps. Marsha BlackburnMarsha BlackburnBlackburn pushes back on potential Corker bid: 'I'm going to win' Nervous GOP seeks new 2018 Senate candidates in three states Corker 'listening closely' to calls to reconsider retirement MORE (R-Tenn.), Dan Burton (R-Ind.), Randy NeugebauerRobert (Randy) Randolph NeugebauerCordray announces he's leaving consumer bureau, promotes aide to deputy director GOP eager for Trump shake-up at consumer bureau Lobbying World MORE (R-Texas), Lamar Smith (R-Texas), Lee Terry (R-Neb.) and Jeb Hensarling (R-Texas).