Kinesin-1 moves processively along microtubules by alternatively moving two motor domains, but the mechanism of the preferential forward stepping is still controversial. The “neck linker-docking model” proposes that the neck linker of the microtubule-bound head propel the tethered head forward. We proposed an alternative “biased-capturing model” that the tethered head freely diffuses and is captured preferentially at the forward-binding site. The latter model predicts that the neck linker of the tethered head, not of the microtubule-bound head, is essential for stepping, but it is difficult to distinguish these effects using “symmetric kinesin dimer”. To distinguish these models, we engineered “asymmetric two-headed monomer” kinesin. We joined two monomer heads tandemly on a single polypeptide, in which the neck linker of first head (Nhead) is connected to second head (Chead) so that it can propel Chead forward, whereas the neck linker of Chead is free and is not connected to Nhead. The neck linker-docking model would predict this mutant could not take the second step. Surprisingly, the two-headed monomer showed robust and unidirectional movement along microtubules in single molecule fluorescent assays. The distance travelled was even longer than wild-type dimer but the velocity was reduced by a factor of 4. Single molecule FRET observation showed that the mutant spent most of the time in a two-head-bound state where the Nhead is leading. These results indicate that the rate-limiting step of the two-headed monomer's processive movement is the forward stepping of Chead driven by the neck linker-docking of Nhead and that the neck linker-docking independent forward stepping of Nhead is rapid and more efficient. These results rule out the idea that neck linker docking is essential to take a forward step and favour the biased-capturing mechanism.