Aims. We probe the dependence of α 2 /μ on the ambient matter density by means of spectral observations in submm-and mm-wave bands. Methods. A procedure is suggested for exploring the value of F = α 2 /μ, where μ = m e /m p is the electron-to-proton mass ratio, and α = e 2 /( c) is the fine-structure constant. The fundamental physical constants, which are measured in different physical environments of high (terrestrial) and low (interstellar) densities of baryonic matter are supposed to vary in chameleon-like scalar field models, which predict that both masses and coupling constant may depend on the local matter density. The parameter ΔF/F = (F obs − F lab )/F lab can be estimated from the radial velocity offset, ΔV = V rot − V fs , between the low-laying rotational transitions in carbon monoxide 13 CO and the fine-structure transitions in atomic carbon [C i]. A model-dependent constraint on Δα/α can be obtained from ΔF/F using Δμ/μ independently measured from the ammonia method. Results. Currently available radio astronomical datasets provide an upper limit on |ΔV| < 110 m s −1 (1σ). When interpreted in terms of the spatial variation of F, this gives |ΔF/F| < 3.7 × 10 −7 . An order of magnitude improvement in this limit will allow us to independently test a non-zero value of Δμ/μ = (2.2 ± 0.4 stat ± 0.3 sys ) × 10 −8 , recently found with the ammonia method. Considering that the ammonia method restricts the spatial variation of μ at the level of |Δμ/μ| ≤ 3 × 10 −8 and assuming that ΔF/F is the same in the entire interstellar medium, one obtains that the spatial variation of α does not exceed the value |Δα/α| < 2 × 10 −7 . Since extragalactic gas clouds have similar densities to those in the interstellar medium, the bound on Δα/α is also expected to be less than 2 × 10 −7 at high redshift if no significant temporal dependence of α is present.