<p>We propose a lattice formulation of four dimensional super Yang-Mills model with a twisted <i>N</i> = 4 supersymmetry in a manifestly gauge covariant manner. The formulation we employ here is a four dimensional extension of the manifestly gauge covariant method which was developed in our proposals of Dirac-Kähler twisted <i>N</i> = <i>D</i> = 2 and <i>N</i> = 4 <i>D</i> = 3 super Yang-Mills on a lattice. Twisted <i>N</i> = 4 supersymmetry algebra is geometrically realized on a four dimensional lattice with link supercharges and the use of link (anti-)commutators. Employing Grassmann parameters with link nature, we explicitly show that the resulting super Yang-Mills action is invariant under all the supercharges on a lattice without chiral fermion problems. As a group and algebraic interpretation of the link approach, we show that promoting bosonic supercovariant derivatives to their exponentials consistently with the lattice Leibniz rule naturally gives rise to the notion of gauge covariant link (anti-)commutators. This can be regarded as a fermionic decomposition of a Lie group element, which may provide a new methodology in super Lie group and super Lie algebra. We also provide a five dimensional lift-up of the formulation with exact <i>N</i> = 4 SUSY invariance on a five dimensional lattice.</p>