A mechanism of extraction of tubular membranes from a lipid vesicle is presented. A concentration gradient of anchoring amphiphilic polymers generates tubes from budlike vesicle protrusions. We explain this mechanism in the framework of the Canham-Helfrich model. The energy profile is analytically calculated and a tube with a fixed length, corresponding to an energy minimum, is obtained in a certain regime of parameters. Further, using a phase-field model, we corroborate these results numerically. We obtain the growth of tubes when a polymer source is added, and the budlike shape after removal of the polymer source, in accordance with recent experimental results.