I assume you either want 1. a fixed endpoint homotopy of curves all having the same start point and the same end point or 2. a free or based homotopy of loops. Otherwise, it would be good to specify what kind of curves you have and what type of homotopy you want.
In either of these situations, the answer is yes. In fact, it holds in a much more general setting.
In a metric space $X$ which has a universal cover (locally path connected, semi-locally simply connected), if $\sigma_l\to\sigma$ uniformly, then there is a $N$ such that $\sigma_l$ is homotopic (in any of the above senses) to $\sigma$ for each $l\geq N$.
Really, you don't even need the metric. If $X$ is path connected, locally path connected, and semi-locally simply connected (the usual ingredients for a universal covering), and you use the compact-open topology on the space of paths (this agrees with the uniform metric when $X$ is metric), then if $\sigma_l\to \sigma$ in this path space, then there is a $N$ such that $\sigma_l$ is homotopic (in any of the above senses) to $\sigma$ for each $l\geq N$.
The general idea is, for large $l$, to connect $\sigma$ and $\sigma_l$ with "small" paths so that you get a sequence of "small" loops between the two. Since small loops in a space with a universal cover are trivial (homotopically), you can piece together all these homotopies to get a homotopy of $\sigma$ and $\sigma_l$.
There is a nice detailed proof of the general statement in
Discreteness and Homogeneity of the Topological Fundamental Group