The present-day thermal state of the martian interior is a very important issue for understanding the 22 internal evolution of the planet. Here, in order to obtain an improved upper limit for the heat flow at 23 the north polar region, we use the lower limit of the effective elastic thickness of the lithosphere loaded 24 by the north polar cap, crustal heat-producing elements (HPE) abundances based on martian geochem- 25 istry, and a temperature-dependent thermal conductivity for the upper mantle. We also perform similar 26 calculations for the south polar region, although uncertainties in lithospheric flexure make the results 27 less robust. Our results show that the present-day surface and sublithospheric heat flows cannot be 28 higher than 19 and 12 mWm2, respectively, in the north polar region, and similar values might be 29 representative of the south polar region (although with a somewhat higher surface heat flow due to 30 the radioactive contribution from a thicker crust). These values, if representative of martian averages, 31 does not necessarily imply sub-chondritic HPE bulk abundances for Mars (as previously suggested), since 32 (1) chondritic composition models produce a present-day total heat power equivalent to an average sur- 33 face heat flow of 14–22 mWm2 and (2) some convective models obtain similar heat flows for the pres- 34 ent time. Regions of low heat flow may even have existed during the last billions of years, in accordance 35 with several surface heat flow estimates of 20 mWm2 or less for terrains loaded during Hesperian or 36 Amazonian times. On the other hand, there are some evidences suggesting the current existence of 37 regions of enhanced heat flow, and therefore average heat flows could be higher than those obtained 38 for the north (and maybe the south) polar region.