  Millisecond pulsars (MSPs) are generally believed to be old neutron stars
(NSs) which have been spun up to high rotation rates via accretion of matter
from a companion star in a low-mass X-ray binary (LMXB). However, many details
of this recycling scenario remain to be understood. Here we investigate binary
evolution in close LMXBs to study the formation of radio MSPs with low-mass
helium white dwarf companions (He WDs) in tight binaries with orbital periods
P_orb = 2-9 hr. In particular, we examine: i) if such observed systems can be
reproduced from theoretical modelling using standard prescriptions of orbital
angular momentum losses (i.e. with respect to the nature and the strength of
magnetic braking), ii) if our computations of the Roche-lobe detachments can
match the observed orbital periods, and iii) if the correlation between WD mass
and orbital period (M_WD, P_orb) is valid for systems with P_orb < 2 days.
Numerical calculations with a detailed stellar evolution code were used to
trace the mass-transfer phase in ~ 400 close LMXB systems with different
initial values of donor star mass, NS mass, orbital period and the so-called
gamma-index of magnetic braking. Subsequently, we followed the orbital and the
interior evolution of the detached low-mass (proto) He WDs, including stages
with residual shell hydrogen burning. We find that a severe fine-tuning is
necessary to reproduce the observed MSPs in tight binaries with He WD
companions of mass < 0.20 M_sun, which suggests that something needs to be
modified or is missing in the standard input physics of LMXB modelling. We
demonstrate that the theoretically calculated (M_WD, P_orb)-relation is in
general also valid for systems with P_orb < 2 days, although with a large
scatter in He WD masses between 0.15-0.20 M_sun. The results of the thermal
evolution of the (proto) He WDs are reported in a follow-up paper (Paper II).
