Context. Early dust grain growth in protostellar envelopes infalling on young disks has been suggested in recent studies, supporting the hypothesis that dust particles start to agglomerate already during the class 0/I phase of young stellar objects. If this early evolution were confirmed, it would impact the usually assumed initial conditions of planet formation, where only particles with sizes ≲0.25 µm are usually considered for protostellar envelopes. Aims. We aim to determine the maximum grain size of the dust population in the envelope of the class 0/I protostar L1527 IRS, located in the Taurus star-forming region (140 pc). Methods. We use Atacama Large millimeter/submillimeter Array and Atacama Compact Array archival data and present new observations, in an effort to both enhance the signal-to-noise ratio of the faint extended continuum emission and properly account for the compact emission from the inner disk. Using observations performed in four wavelength bands and extending the spatial range of previous studies, we aim to place tight constraints on the spectral (α) and dust emissivity (β) indices in the envelope of L1527 IRS. Results. We find a rather flat α ~ 3.0 profile in the range 50–2000 au. Accounting for the envelope temperature profile, we derived values for the dust emissivity index, 0.9 < β < 1.6, and reveal a tentative, positive outward gradient. This could be interpreted as a distribution of mainly interstellar medium like grains at 2000 au, gradually progressing to (sub)millimeter-sized dust grains in the inner envelope, where at R = 300 au, β = 1.1 ± 0.1. Our study supports a variation of the dust properties in the envelope of L1527 IRS. We discuss how this can be the result of in situ grain growth, dust differential collapse from the parent core, or upward transport of disk large grains.