Such an ideal does not exist. 

Indeed, suppose the contrary, and let $I$ be such an ideal. The sequence $(x_n)=(1,0,1,0,\dots)$ is [almost convergent][1], and therefore $I$-convergent, to $1/2$. So, 
$$\mathbb N=\{n\in\mathbb N\colon|x_n-1/2|\ge1/2\}\in I, 
$$
and hence $I$ is the powerset of $\mathbb N$. So, every sequence is $I$-convergent, and therefore almost convergent, to every real limit, which is of course absurd. 


  [1]: https://en.wikipedia.org/wiki/Banach_limit#Almost_convergence