In the present work, we propose to investigate the production of (d_{N Omega }) in the (Omega ^{-} d rightarrow p d_{N Omega }^-) process by utilizing an effective Lagrangian approach, where (d_{N Omega }) is identified as (NOmega ) bound state with the binding energy (E_{b}=2.46) MeV. Experimentally, the J-PARC hadron facility proposed to investigate the (K^{-}p rightarrow Omega ^{-} bar{K}^{(*)0} K^{+}) process, which is expected to yield an (Omega ) beam with the momentum of approximately 3 GeV. Additionally, theoretical studies of the (psi (2S) rightarrow Omega ^{-} bar{Omega }^{+}) process at BESIII provided an (Omega ) beam with the momentum of 774 MeV. Considering these two potential (Omega ) beam sources, our estimations show that for the (Omega ^{-} d rightarrow p d_{N Omega }^-) process, the cross sections are (Big (329.7^{+26.9}_{-49.6}Big )) (mu )b, (Big (174.0^{+26.5}_{-38.2}Big )) (mu )b, (Big (16.9^{+7.4}_{-7.7}Big )) (mu )b, and (Big (2.0^{+1.8}_{-1.4}Big )) (mu )b at (P_{Omega } =) 0.7, 0.9, 2.0, and 4.0 GeV, respectively, where the central values are estimated with (Lambda _{r}=1.0) GeV, and the errors come from the variation of (Lambda _{r}) from 0.8 to 1.2 GeV. We also estimate the differential cross sections, which reach the maximum at the forward angle limit. In addition, since the (d_{N Omega }) dibaryon predominantly decays into (Xi Lambda ). Therefore, we further investigate the (Omega ^{-} d rightarrow p Xi ^- Lambda ) process and estimate the relevant cross sections. It is expected that the present estimations can be tested by further experimental measurements at J-PARC and STCF in the future.
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