2. Given three positive integers $m$, $n$, and $k$ satisfying $m+n=$ $2021, m-3k$, and $n+k$ are both prime numbers. Then $m+2n+3k=$
2. 2025 or 4040.

If $n+k=2$, then $n=1, k=1$.
Thus, $m=2020$.
At this point, $m-3k=2017$ (prime number), which meets the condition.
Therefore, $m+2n+3k=2025$.
If $n+k>2$, then $n+k$ is an odd number, and
$$
(m-3k)+(n+k)=2021-2k
$$

is an odd number, so $m-3k$ is even.
Since $m-3k$ is a prime number, thus,
$$
m-3k=2 \Rightarrow 3k=m-2 \text{. }
$$

At this point, when $m=5, k=1, n=2016$, $n+k$ is the prime number 2017.
$$
\begin{array}{l}
\text{ Hence } m+2n+3k=2(m+n)-2 \\
=2 \times 2021-2=4040 .
\end{array}
$$

In summary, the value of $m+2n+3k$ is 2025 or 4040.