Abstract: Galaxy clusters might be sources of TeV gamma rays emitted by high-energy protons and electrons accelerated by large scale structure formation shocks, galactic winds, or active galactic nuclei. Furthermore, gamma rays may be produced in dark matter particle annihilation processes at the cluster cores. We report on observations of the galaxy clusters Perseus and Abell 2029 using the 10 m Whipple Cherenkov telescope during the 2003-2004 and 2004-2005 observing seasons. We apply a two-dimensional analysis technique to scrutinize the clusters for TeV emission. In this paper we first determine flux upper limits on TeV gamma-ray emission from point sources within the clusters. Second, we derive upper limits on the extended cluster emission. We subsequently compare the flux upper limits with EGRET upper limits at 100 MeV and theoretical models. Assuming that the gamma-ray surface brightness profile mimics that of the thermal X-ray emission and that the spectrum of cluster cosmic rays extends all the way from thermal energies to multi-TeV energies with a differential spectral index of -2.1, our results imply that the cosmic ray proton energy density is less than 7.9% of the thermal energy density for the Perseus cluster.