Global net-negative carbon emissions are prevalent in almost all emission pathways that meet the Paris temperature targets. In this paper, we generate and compare cost-effective emission pathways that satisfy two different types of climate targets. First, the common approach of a radiative forcing target that has to be met by the year 2100 (RF2100), and, second, a temperature ceiling target that has to be met over the entire period, avoiding any overshoot. Across two integrated assessment models (IAMs), we found that the amount of net-negative emissions—when global net emissions fall below zero—depends to a large extent on how the target is represented, i.e. implemented in the model. With a temperature ceiling (no temperature overshoot), net-negative emissions are limited and primarily a consequence of trade-offs with non-CO2 emissions, whereas net-negative emissions are significant for the RF2100 target (temperature overshoot). The difference becomes more pronounced with more stringent climate targets. This has important implications: more stringent near-term emission reductions are needed when a temperature ceiling is implemented compared to when an RF2100 target is implemented. Further, in one IAM, for our base case assumptions, the cost-effective negative carbon emissions (i.e. gross anthropogenic removals) do not depend to any significant extent on how the constraint is implemented, only, largely, on the ultimate stringency of the constraint. Hence, for a given climate target stringency in 2100, the RF2100 target and the temperature ceiling may result in essentially the same amount of negative carbon emissions. Finally, it is important that IAM demonstrate results for diverse ways of implementing a climate target, since the implementation has implications for the level of near-term emissions and the perceived need for net-negative emissions (beyond 2050).