  We introduce the so-called Bishop-Phelps-Bollob\'as property for positive
functionals, a particular case of the Bishop-Phelps-Bollob\'as property for
positive operators. First we show a version of the Bishop-Phelps-Bollob\'as
theorem for positive elements and positive functionals in the dual of any
Banach lattice. We also characterize the strong Bishop-Phelps-Bollob\'as
property for positive functionals in a Banach lattice. We prove that any
finite-dimensional Banach lattice has the the Bishop-Phelps-Bollob\'as property
for positive functionals. A sufficient and a necessary condition to have the
Bishop-Phelps-Bollob\'as property for positive functionals are also provided.
As a consequence of this result, we obtain that the spaces $L_p(\mu)$ ($1\le p
< \infty$), for any positive measure $\mu,$ $C(K)$ and $\mathcal{M} (K)$, for
any compact and Hausdorff topological space $K,$ satisfy the
Bishop-Phelps-Bollob\'as property for positive functionals. We also provide
some more clarifying examples.
