This paper is the first in a series of three with the aim of addressing one of the controversial issues at the magnetopause, namely the location where reconnection first occurs during periods of a large interplanetary magnetic field By. In this first paper, the linear properties of the collisionless tearing mode are reexamined as a function of the guide field By using a formally exact approach for computing the nonlocal Vlasov stability of a current layer. Three distinct parameter regimes are identified depending on the degree to which electron orbits are modified by the guide field in the central region of the current layer. In the limit of both weak and strong guide field, the fastest-growing tearing mode has a wave vector kx perpendicular to the direction of the current, in agreement with previous theoretical treatments. However, for intermediate values of the guide field where the electrons begin to transition to magnetized orbits, the fastest-growing modes have a finite wave vector ky in the direction of the current. In this newly discovered regime, the so-called drift tearing modes have finite real frequency and propagate in the direction of the electron diamagnetic drift with growth rates 10--50% larger than the conventional tearing instability. Maximum growth occurs for a propagation angle in the range $theta$ = tan-1(ky/kx) ≈ 6--10¿. These new predictions are confirmed using fully kinetic particle-in-cell simulations. The structure of the out-of-plane magnetic field perturbation predicted by nonlocal Vlasov theory is examined as a function of guide field. In the limit of a neutral sheet, the quadrupole structure has a characteristic scale near the electron meandering width and shows significant differences with the predictions of linear Hall MHD. The addition of a guide field strongly distorts the quadrupole structure and compresses the spatial extent. In the strong guide field limit, the width of the out-of-plane magnetic field perturbation is reduced to the electron gyroscale in the guide field. During the onset phase, these structures represent a distinct signature of the collisionless tearing mode that is significantly different than the typical ion-scale quadrupole pattern from fast reconnection. Finally, we note that the tearing mode maintains a significant growth rate over a large range of guide field so that component merging cannotbe ruled out based on linear theory.