The time dependences, up to 200 s, and the noise power spectrum (0.005-10 Hz) in the electron transport response at bias up to 300 mV of Co/Al₂O₃/Ni₈₀Fe₂₀ magnetic tunnel junctions (MTJs) and of Co/Al₂O₃δ(Fe)/Ni₈₀Fe₂₀ (with Fe δ dopants of thickness 1.8 Å inside the barrier) were investigated. The magnetic field was changed between +100 and -100 G in steps of 1 G. The measurements were carried out at different temperatures between 77 and 300 K for the samples with large tunnel magnetoresistance (exceeding 14% at 300 K). We found that the magnetization reversal of the Co and permalloy electrodes, as detected from the time response near the coercive field, occurs via relaxation on the timescale of about 10² s with sudden jumps in the resistance (ΔR/R~10^-2-10^-3). We link this noise to the depinning of the domain walls. In addition to the magnetic noise, in some of the studied MTJs with Fe δ dopants, we observed a two-level-system telegraph-type noise, which was independent of the magnetic field, indicating its relation to the trapped charges inside the insulating barrier. For MTJs, the noise power spectrum has 1/f^α character for a wide frequency range below a few Hz. At low bias and parallel state the exponent α is close to 1-1.5, but at higher bias or in the antiparallel state the exponent increases to 2. We link these effects to non-equilibrium noise in magnetic structure of the electrodes in the antiparallel state and to non-equilibrium transport inside the barrier at high bias.