In this Phase I STTR program, we propose to develop a conceptual design of a 40-T, ?2-cm-warm- bore superconducting solenoid magnet for use in high energy particle physics experiments, e.g., muon ionization cooling, by combining a Low Temperature Superconducting background magnet and a High Temperature Superconducting insert magnet. High Temperature Superconductors must be employed for very-high-field (>25 T) superconducting magnets because of its exceptional in- field current-carrying performance compared with conventional Low Temperature Superconductors. Recently available, high quality REBCO tape, with greater mechanical strength than other types of High Temperature Superconducting (Bi2223, Bi2212) tapes and wires, is the most suitable conductor for our proposed insert coil set of the 40-T magnet system. The aims of Phase I are to: 1) perform a conceptual design of a 26-T High Temperature Superconducting insert coil and a 14-T Low Temperature Superconducting background magnet to be operated at 40 T and 4.2 K; 2) analyze and design a quench protection system for the 40-T magnet system; 3) perform lab-scale tests to validate analyses and design concepts. For high-field REBCO magnets, the most severe design limiting parameters are critical current and internal stresses induced by enormous electromagnetic forces. Key program elements include: 1) divide a High Temperature Superconducting insert into two nested coils to reduce both hoop and radial stresses; 2) evaluate eliminating the turn-to-turn insulation layers using a type of No- Insulation winding technique; (3) adopt multi-width, i.e., graded critical current, REBCO windings co-wound with stainless steel tapes; 4) investigate, analytically and experimentally, the screening- current-induced stress; and 5) adopt multiple strategies including a detect-and-activate-heater method to minimize risk of damage from a magnet quench; (5) Other conductor and winding concepts will also be evaluated such as insulated high current multi-tape cable conductors. We believe that our design for a 40-T hybrid solenoid magnet will become an enabling technology, not only for high energy particle physics applications, but also for high-field Nuclear Magnetic Resonance magnets for drug discovery/development, and for high-field research magnets required for condensed matter studies. In Phase II of this project, we will design, build, and test a High Temperature Superconducting prototype coil to validate the design concepts.