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We provide promising computational (in silico) data on phytochemicals (compounds <b>1</b>–<b>10</b>) from Arabian Peninsula medicinal plants as strong binders, targeting 3-chymotrypsin-like protease (3CL^Pro) and papain-like proteases (PL^Pro) of SARS-CoV-2. Compounds <b>1</b>–<b>10</b> followed the Lipinski rules of five (RO5) and ADMET analysis, exhibiting drug-like characters. Non-covalent (reversible) docking of compounds <b>1</b>–<b>10</b> demonstrated their binding with the catalytic dyad (CYS145 and HIS41) of 3CL^Pro and catalytic triad (CYS111, HIS272, and ASP286) of PL^Pro. Moreover, the implementation of the covalent (irreversible) docking protocol revealed that only compounds <b>7</b>, <b>8</b>, and <b>9</b> possess covalent warheads, which allowed the formation of the covalent bond with the catalytic dyad (CYS145) in 3CL^Pro and the catalytic triad (CYS111) in PL^Pro. Root-mean-square deviation (RMSD), root-mean-square fluctuation (RMSF), and radius of gyration (Rg) analysis from molecular dynamic (MD) simulations revealed that complexation between ligands (compounds <b>7</b>, <b>8</b>, and <b>9</b>) and 3CL^Pro and PL^Pro was stable, and there was less deviation of ligands. Overall, the in silico data on the inherent properties of the above phytochemicals unravel the fact that they can act as reversible inhibitors for 3CL^Pro and PL^Pro. Moreover, compounds <b>7</b>, <b>8</b>, and <b>9</b> also showed their novel properties to inhibit dual targets by irreversible inhibition, indicating their effectiveness for possibly developing future drugs against SARS-CoV-2. Nonetheless, to confirm the theoretical findings here, the effectiveness of the above compounds as inhibitors of 3CL^Pro and PL^Pro warrants future investigations using suitable in vitro and in vivo tests.