A finite reductive group G^F is defined as the fixed points of a reductive group G/F_q under the q-Frobenius endomorphism. Their representations were studied by Deligne-Lusztig and Brokemper gave a description of their intersection ring. Focusing on the latter, we want to relate Tate motives on F_q with G-action to equivariant Tate motives on the associated flag variety. As Tate motives admit, in certain cases, a t-structure, this should lead to a first access point of motivic representation theory of finite reductive groups.