Spherical Tokamaks (STs) with a high normalised plasma pressure parameter βT are promising alternative pathways for net gain magnetic fusion reactors. Toroidal Alfvén eigenmodes (TAEs) excited via wave-particle resonances by fusion-born alpha-particles are of major concern for burning deuterium-tritium (D-T) plasmas in magnetic fusion as AEs cause a radial re-distribution of alpha-particles affecting their plasma heating efficiency and causing alpha-losses to the first wall. Tokamak plasmas with high pressure gradients were found to suppress toroidally induced Alfvén eigenmodes (TAEs) in ideal incompressible MHD and the high plasma βT attained by STs is expected to facilitate access to these TAE-free regimes. However, the low magnetic shear inherent to STs can lead to the formation of multiple AEs, which may increase transport due to the larger resonance area of the alpha-particle phase space. A suite of MHD codes HELENA, MISHKA and CS_MISH are used to investigate the discrete AE spectrum in high-pressure STs in the current paper. TAEs for an experimental ST40 magnetic equilibrium are computed and through varying βT and the current density profile the suppression of TAEs is investigated. The critical βT (above which no TAE modes are found) is shown to decrease with increased ellipticity, increased toroidal field (relative to a fixed plasma current) and a flatter current density profile. Despite the increase in the number of core-localised TAEs at low magnetic shear, the modes appear more sensitive to increases in βT and disappear at lower thresholds.