What if a council is interested in proposing an amendment and voters are interested in proposing a different amendment on the same subject. Do the statutes deal with competing proposals?

Somewhat, although the statutory provisions are incomplete and perhaps ambiguous.

The fullest treatment is in G.S. 160A-104, which deals with voter-initiated proposals. It prohibits voters from filing a petition with the city in three situations:

1. When the council has initiated its own proposal and has published the notice of a public hearing on that proposal. This bar continues until the council has either adopted or rejected an ordinance on its proposal or, if the council takes no action, until the time for taking action has passed. (Recall that the council must act within 60 days after the public hearing.)
2. When the council has adopted an ordinance amending the charter – whether the result of a council-initiated proposal or a citizen-initiated proposal. The section prohibits filing a petition within 18 months following the effective date of the ordinance.
3. When the voters in a referendum have rejected proposed charter amendments – whether proposed by the council or by citizens. The section prohibits filing a petition for 18 months after the date of the referendum.

It should be stressed that the statute specifies that these prohibitions apply only to petitions that propose amendments on the same subject matter as the council proceeding, or adopted ordinance, or defeated referendum. There is a parallel but much less complete treatment in G.S. 160A-102, which deals with council-initiated proposals. That section bars council action to begin the amendment process once the city has received a petition proposing charter amendments, but it says nothing about limits on council action when an earlier ordinance has become effective or when the voters have defeated a proposal in a referendum. The provision in G.S. 160A-102 also does not limit its reach to amendments on the same subject matter, but perhaps such a limitation should be implied. It is hard to understand why receipt of a petition to increase council size or change council terms should bar a council-initiated proposal to change to the council-manager system or to give the mayor the right to vote on all issues.