The short answer is: no, not without incurring an exponential cost, if by "block" you mean a block in the middle of a wavefunction represented by an MPS, versus a block starting from site 1 and continuing through some site j.
The longer answer is that it's quite likely there is an efficient, approximate method that works well in practice for obtaining the entropy of a large block in the middle of an MPS. A simple example of such a method would be to swap the sites j,...,k with the sites 1,...,j through a set of local swap operations until all of these sites are exchanged with each other. Then one could compute the entanglement of the resulting MPS via a cut at bond j,j+1 and it would approximately equal the block entanglement of the original state. But the key question in this procedure is whether all of those sites can be efficiently swapped without leading to a new bond dimension that's too huge.
There might be smarter ways to do something similar, using a tree tensor network maybe, or some other methods that are yet to be explored (randomized SVD, etc.). It's sort of an open, under explored topic.
Why do you need the entanglement of a large block not starting from one edge of the system?