We consider two-dimensional directed polymers in random environment in the sub-critical regime and in the quasi-critical regime introduced recently by Caravenna, Cottini and Rossi, arXiv:2307.02453v1. For $q\leq q_N$ with $q_N\to\infty$ diverging at a suitable rate with the size of the system, we obtain upper bound estimates on the $q$-moment of the partition function for general environments. In the sub-critical regime, our results improve the $q_N$-threshold obtained for Gaussian environment in Cosco, Zeitouni, Comm. Math. Phys (2023). As a corollary, we derive large deviation estimates with a Gaussian rate function.