Message Passing Interface (MPI) is a standard library interface for writing parallel programs. The MPI specification is broadly used for solving engineering and scientific problems on parallel computers, and MPICH2 is a popular MPI implementation developed at Argonne National Laboratory. The scalability of MPI implementations is very important for building high performance parallel computing applications. The initial TCP (Transmission Control Protocol) network module developed for Nemesis communication sub-system in the MPICH2 library, however, was not scalable in how it established connections: pairwise connections between all of an application's processes were established during the initialization of the application (the library call to MPI_Init), regardless of whether the connections were eventually needed or not. In this work, we have developed a new TCP network module for Nemesis that establishes connections on-demand. The on-demand connection establishment scheme is designed to improve the scalability of the TCP network module in MPICH2 library, aiming to reduce the initialization time and the use of operating system resources of MPI applications. Our performance benchmark results show that MPI_Init in the on-demand connection establishment scheme becomes a fast constant time operation, and the additional cost of establishing connections later is negligible. The on-demand connection establishment between two processes, especially when two processes attempt to connect to each other simultaneously, is a complex task due to race-conditions and thus prone to hard-to-reproduce defects. To assure ourselves of the correctness of the TCP network module, we modeled its design using the SPIN model checker, and verified safety and liveness properties stated as Linear Temporal Logic claims.