They discussed the proposal to impose limits at a regional summit.
Togo and The Gambia, both with presidents who have been in power for more than two terms, opposed the idea, diplomats say.
The third-term issue has caused a lot of controversy in several African states in recent years.
Burkina Faso’s President Blaise Compaore was forced out of office last year after trying to change the constitution so he could run for a third term.
In East Africa, there have been weeks of protests and a failed coup after Burundi’s President Pierre Nkurunziza announced plans to seek a third term in office.’
The constitutions of most West African states already impose a two-term limit.
The proposal discussed at Tuesday’s summit of the regional body Ecowas in the Ghanaian capital, Accra, was supposed to formalise this across the region.
If a popular referendum on presidential term limits was held across the region, there is little doubt that people would vote in favour of it.
Since the president of Burkina Faso was overthrown last year for trying to change the constitution so he could seek re-election, this has become a big regional issue.
However, some West African leaders argue that each country has a different political context and there cannot be a blanket ruling.
Optimistic souls would say that it is already a step forward that presidential terms are being discussed at an Ecowas summit.
But imposing a two-term limit limit requires unanimity among West African leaders, and there is still a long way to go to achieve that.
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