Let $x_i:\Lambda_i\to X_i$ be a net in a topological spaces $X_i$ that converges to $a_i\in X_i$ where $i\in\{1,2\}$
Does there exist a poset $\Lambda$ and subnets of $x_i:\Lambda_i\to X_i$ of the form $y_i:\Lambda\to X_i$ ?
You can take a look at this notes to remind definitions of net and especially of a subnet.
There's the simple construction $\Lambda = \Lambda_1 \times \Lambda_2$ with the partial order
$$(\alpha,\beta) \leqslant (\gamma,\delta) \iff \alpha \leqslant \gamma \land \beta \leqslant \delta,$$
and $y_i = x_i \circ \pi_i$, where $\pi_i$ is the projection onto the $i$-th component.
In a given situation with known $\Lambda_i$, it is likely that one can construct more interesting index sets and subnets, but without any specific data, I don't see a way to produce something interesting.