Council members David Ryu and Nury Martinez introduced the policy at Tuesday's Los Angeles City Council meeting.
The council members said the state law allows for up to 18 weeks of paid family leave but only provides eligible employees with 60 to 70 percent of their wages up to a state maximum.
The proposed city program would seek to augment the state's contribution via the existing State Disability Insurance and Paid Parental Leave programs to cover 100 percent of wages.
"We know in Los Angeles, 60 to 70 percent of your pay is not enough to pay the bills, let alone to start a family,'' Ryu said at a news conference at City Hall.
The program would be paid by the employers, but could create an exemption or risk pool for small businesses and nonprofits, the council members said.
"By providing a full, paid parental leave, we are going to empower Los Angeles' hard-working mothers and fathers to be outstanding parents without fear that they are going to lose their job or lose a paycheck,'' Martinez said.
Gov. Gavin Newsom has proposed a six-month paid family leave policy, but the idea is still in the early stages and the governor is forming a task force on the topic.
Los Angeles Mayor Eric Garcetti said he supports the proposal, thanking the two councilmembers for "leading the way...to make paid parental leave a reality for Angelenos."
Nearly every other country in the developed world guarantees paid family leave.— Mayor Eric Garcetti (@MayorOfLA) January 29, 2019
I applaud Councilmembers @DavideRyu and @CD6Nury for leading the way, and look forward to working with them to make paid parental leave a reality for Angelenos.
Statement: https://t.co/1IgQbim9lM pic.twitter.com/lc9ryILWmL
City News Service contributed to this report.