Context. Lyman break analogs in the local Universe serve as counterparts to Lyman break galaxies (LBGs) at high redshifts, which are widely regarded as major contributors to cosmic reionization in the early stages of the Universe. Aims. We studied XMM-Newton and Chandra observations of the nearby LBG analog Haro 11, which contains two X-ray-bright sources, X1 and X2. Both sources exhibit Lyman continuum (LyC) leakage, particularly X2. Methods. We analyzed the X-ray variability using principal component analysis (PCA) and performed spectral modeling of the X1 and X2 observations made with the Chandra ACIS-S instrument. Results. The PCA component, which contributes to the X-ray variability, is apparently associated with variable emission features, likely from ionized superwinds. Our spectral analysis of the Chandra data indicates that the fainter X-ray source, X2 (X-ray luminosity LX ∼ 4 × 1040 erg s- 1), the one with higher LyC leakage, has a much lower absorbing column (NH ∼ 1.2 × 1021 cm- 2) than the heavily absorbed luminous source X1 (LX ∼ 9 × 1040 erg s- 1 and NH ∼ 11.5 × 1021 cm- 2). Conclusions. We conclude that X2 is likely less covered by absorbing material, which may be a result of powerful superwinds clearing galactic channels and facilitating the escape of LyC radiation. Much deeper X-ray observations are required to validate the presence of potential superwinds and determine their implications for the LyC escape.