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The recent discovery of the quantum nonlinear Hall effect has revived the field of nonlinear transport. Here, we investigate the magnetic-field-induced nonlinear transport in time reversal symmetric Weyl semimetals. We show that the interplay of the band-geometric quantities, such as the Berry curvature, and the magnetic part of the Lorentz force can give rise to finite nonlinear Hall conductivity that is linear in the magnetic field. In addition, we show that the chiral chemical potential which represents chiral anomaly gives rise to linear magnetic-field-dependent nonlinear longitudinal conductivities along with the nonlinear Hall conductivities. Such nonlinear conductivities can manifest through nonlinear transport measurement as well as nonlinear optical phenomena like photocurrent and the second harmonic generation.