We evaluate the discovery potential for the heavy Higgs bosons at the LHC energy upgrade with TeV. We assume the degenerate mass spectrum and an approximate alignment limit in the Type-II Two Higgs Doublet Model for illustration. We explore the observability of the heavy neutral Higgs bosons by examining the clean signals from via gluon-gluon fusion production. The associated production of a top quark and a charged Higgs boson via is adopted to predict the discovery potential of heavy charged Higgses. We also emphasize the potential importance of the electroweak production of Higgs boson pairs, i.e. and . These are only governed by pure electroweak gauge couplings and can provide complementary information to the conventional signals in the determination of the nature of the Higgs sector.