We obtain a large class of AdS spacetimes warped with certain internal spaces in 11-dimensional and type IIA or IIB supergravities. The warp factors depend only on the internal coordinates. These solutions arise as the near-horizon geometries of more general semilocalized multi-intersections of p-branes. We achieve this by noting that any sphere (or AdS spacetime) of dimension greater than 3 can be viewed as a foliation involving ${\mathrm{S}}^{3}$ (or ${\mathrm{AdS}}_{3}).$ Then the ${\mathrm{S}}^{3}$ (or ${\mathrm{AdS}}_{3})$ can be replaced by a three-dimensional lens space (or a BTZ black hole), which arises naturally from the introduction of a NUT (or a $\mathrm{pp}$ wave) to the M-branes or the D3-brane. We then obtain multi-intersections by performing a Kaluza-Klein reduction or Hopf T-duality transformation on the fiber coordinate of the lens space (or the BTZ black hole). These geometries provide further possible examples of the AdS/CFT correspondence and of consistent embeddings of lower-dimensional gauged supergravities in $D=11$ or $D=10.$