Suppose that we have a multivariate normal random vector
$$
(\log X_1,\dots,\log X_k) \sim N(\mu,\Sigma) \, ,
$$
with $\mu\in\mathbb{R}^k$ and $k\times k$ full rank symmetric positive definite matrix $\Sigma=(\sigma_{ij})$.
For the lognormal $(X_1,\dots,X_k)$ it is not difficult to prove that
$$
m_i := \textrm{E}[X_i] = e^{\mu_i + \sigma_{ii}/2} \, , \quad i=1,\dots,k\, ,
$$
$$
c_{ij} := \textrm{Cov}[X_i,X_j] = m_i \,m_j \,(e^{\sigma_{ij}} - 1) \, , \quad i,j=1,\dots,k\, ,
$$
and it follows that $c_{ij}>-m_im_j$.
Hence, we can ask the converse question: given $m=(m_1,\dots,m_k)\in\mathbb{R}^k_+$ and $k\times k$ symmetric positive definite matrix $C=(c_{ij})$, satisfying $c_{ij}>-m_im_j$, if we let
$$
\mu_i = \log m_i - \frac{1}{2} \log\left(\frac{c_{ii}}{m_i^2} + 1 \right) \, , \quad i=1,\dots,k \, ,
$$
$$
\sigma_{ij} = \log\left(\frac{c_{ij}}{m_i m_j} + 1 \right) \, , \quad i,j=1,\dots,k \, ,
$$
we will have a lognormal vector with the prescribed means and covariances.
The constraint on $C$ and $m$ is equivalent to the natural condition $\textrm{E}[X_i X_j]>0$.