He did state publicly that he promised NASA that he could build a rocket comparable to the SLS on a fixed-price $2 billion contract (meaning NASA would not pay a dime for budget overruns), although that price didn't include any second-stage upgrades NASA might require to meet its needs.
SpaceX is actually going ahead with their SLS-like competitor (Codenamed "BFR", I think you can guess what that stands for), and they're supposed to start testing on the methane-powered engines (Raptor) soon, which are supposed to be both more powerful and more efficient than the F-1 engines used in the Saturn V. However, without any customers paying for the R&D, BFR will take a lot longer to build than it would have if NASA contracted SpaceX to do it.
So, yeah. SpaceX offered NASA a contract to build an entire replacement for the SLS for less than a year of SLS funding.