According to May, an operad $\mathcal{C}$ valued in sets is equivalent to a monad in Cat on the endofunctor $C\colon X\mapsto \coprod_i \mathcal{C}(i)\times X^i.$
According to Leinster, an operad is equivalent to a monad in the bicategory of modified spans, diagrams of the form $T1\leftarrow C\to 1,$ where $T$ is the free monoid functor.
Is there some connection between these two monads, other than the fact that they unravel in base cases to the definition of an operad?
Well the two monads are quite different: in May definition you deal with an actual monad in $\mathbf {Cat}$ (i.e. a strict-$2$-category) while in the second case you work with monads in the bicategory of sets and spans in $\mathbf {Set}$ (actually this second kind of monad is a $T$-operad).
Nonetheless the two monads are actually linked together: May's monad is the monad functor part used to build the algebras of Leinster's $T$-operad. Here follows the details of the construction.
Working a little bit with Leinster definition of operads you can see that the span diagram $T1 \leftarrow C \rightarrow 1$ is characterized completely by the left arrow $p \colon C \to T1\cong \mathbb N$ and such arrow can be seen as the family of elements $$C_{i}=\{ c \in C \mid p(c)=i \in T1\}$$ (that's basically an application of the equivalence between fibrations in $\mathbf{Set}$ and indexed families of sets).
Then the functor $\mathcal C \colon \mathbf {Set} \to \mathbf {Set}$ that gives May's monad can be recoverd as
$$\mathcal C(X) = \bigsqcup_{i \in T1} C_i \times X^i$$
If your reference for $T$-operads is Leinster's Higher Operads Higher Categories then you should recognize that this is the monad induced by a $T$-operad whose algebras are the algebras of the $T$-operads.
