It is important to induce and manipulate magnetism in two-dimensional materials for developing low-dimensional spintronic devices. Here we explored the modulation of magnetism in MoS2 monolayers doped with 3d and 4d transition-metal (TM) atoms by strain engineering using first-principles calculations. It is found that 3d TM atom doping tends to induce high spin in TM-MoS2 monolayers, while 4d TM atom doping tends to induce low spin. The magnetic states in TM-MoS2 monolayers can be further mediated by TM-S bond interaction and change of crystal structure symmetry, which is achieved by strain engineering. An increase in the ionic bonding interaction of TM-S modulated by strain leads to an increase of the unpaired electrons accumulated on the TM and S atoms, thus giving rise to the interesting variation in the magnetic moments with strain. Strain is also able to modulate the crystal structures to undergo or not Jahn-Teller (J-T) distortion effects. The structures having J-T distortion will result in low spin states. The research results offer important theoretical support for further application of strain-driven spin devices on MoS2 nanostructures.