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 BlackburnGoogle will no longer use data from personal Gmail accounts for advertising Overnight Regulation: Labor groups fear rollback of Obama worker protection rule | Trump regs czar advances in Senate | New FCC enforcement chief Overnight Tech: Uber CEO resigns | Trump's Iowa tech trip | Dems push Sessions to block AT&T-Time Warner deal | Lawmakers warned on threat to election systems | MORE (R-Tenn.), Dan Burton (R-Ind.), Randy NeugebauerRandy NeugebauerWarren’s regulatory beast is under fire – and rightfully so Dem senators to Trump: Don't tell consumer bureau chief 'you're fired' Overnight Finance: Carson, Warren battle at hearing | Rumored consumer bureau pick meets Trump | Trump takes credit for Amazon hirings | A big loss for Soros MORE (R-Texas), Lamar Smith (R-Texas), Lee Terry (R-Neb.) and Jeb Hensarling (R-Texas).