This might be a naive answer but here is a suggestion for the definition of $\pi_n$-fibrations: maps having the RLP with respect to the the map $\Delta^k\to\Delta^k\times I$ for any $\leq n$. 

In the limit you will get the "obvious" fibrant-object structure on $Top$ which comes from its usual model structure (recall that the full subcategory of fibrant objects in any model category is a category of fibrant objects.. and that all objects are fibrants in $Top$.