WASHINGTON—Senate Republicans began a frenzied week of negotiations to pass a landmark tax overhaul, grappling with several blocs of wavering GOP senators and trying to cobble together enough votes.

One group, including Ron Johnson (R., Wis.) and Steve Daines (R., Mont.), wants deeper tax cuts for so-called pass-through businesses such as partnerships and S corporations that pay taxes on individual rather than corporate tax returns. Both said they want to prevent large corporations from deducting state and local taxes, freeing...