Recent results reported by the ATLAS and CMS experiments on the search for a SM-like Higgs boson both show an excess for a Higgs mass near 125 GeV, which is mainly driven by the γγ and ZZ⁎ decay channels, but also receives some support from channels with a lower mass resolution. We discuss the implications of this possible signal within the context of the minimal supersymmetric Standard Model (MSSM), taking into account previous limits from Higgs searches at LEP, the Tevatron and the LHC. The consequences for the remaining MSSM parameter space are investigated. Under the assumption of a Higgs signal we derive new lower bounds on the tree-level parameters of the MSSM Higgs sector. We also discuss briefly an alternative interpretation of the excess in terms of the heavy CP-even Higgs boson, a scenario which is found to be still viable.