We study the homogeneous periodic XXX Heisenberg spin-$1/2$ chain further deformed by the Jordanian $sl(2)$ twist element. Namely, we show how the Jordanian twist element deforms the projector operators of the spectral decomposition of the $SL(2)$-invariant Yang $R$-matrix. We explicitly express the monodromy matrix entries in terms of the local spin operators and obtain the creation operators of the Bethe vectors as the complicated polynomials of the entries of the monodromy matrix. Moreover, we show that the off-shell action of the transfer matrix on the Bethe vectors has the expected triangular form.