Abstract Vascularization is considered to be one of the key challenges in engineering functional 3D tissues. Engineering suturable vascular grafts containing pores with diameter of several tens of microns in tissue engineered constructs may provide an instantaneous blood perfusion through the grafts improving cell infiltration and thus, allowing rapid vascularization and vascular branching. The aim of this work was to develop suturable tubular scaffolds to be integrated in biofabricated constructs, enabling the direct connection of the biofabricated construct with the host blood stream, providing an immediate blood flow inside the construct. Here, tubular grafts with customizable shapes (tubes, Y-shape capillaries) and controlled diameter ranging from several hundreds of microns to few mm are fabricated based on poly(glycerol sebacate) (PGS)/poly(vinyl alcohol) (PVA) electrospun scaffolds. Furthermore, a network of pore channels of diameter in the order of 100 μm was machined by laser femtosecond ablation in the tube wall. Both non-machined and laser machined tubular scaffolds elongated more than 100% of their original size have shown suture retention, being 5.85 and 3.96 N mm⁻² respectively. To demonstrate the potential of application, the laser machined porous grafts were embedded in gelatin methacryloyl (GelMA) hydrogels, resulting in elastomeric porous tubular graft/GelMA 3D constructs. These constructs were then co-seeded with osteoblast-like cells (MG-63) at the external side of the graft and human umbilical vein endothelial cells inside, forming a bone osteon model. The laser machined pore network allowed an immediate endothelial cell flow towards the osteoblasts enabling the osteoblasts and endothelial cells to interact and form 3D structures. This rapid vascularization approach could be applied, not only for bone tissue regeneration, but also for a variety of tissues and organs.