  Wormhole solutions in General Relativity (GR) require \textit{exotic} matter
sources that violate the null energy condition (NEC), and it is well known that
higher-order modifications of GR and some alternative matter sources can
support wormholes. In this study, we explore the possibility of formulating
traversable wormholes in $f(R)$ modified gravity, which is perhaps the most
widely discussed modification of GR, with two approaches. First, to investigate
the effects of geometrical constraints on the global characteristics, we gauge
the $rr$-component of the metric tensor, and employ Pad\`{e} approximation to
check whether a well-constrained \textit{shape function} can be formulated in
this manner. We then derive the field equations with a background of string
cloud, and numerically analyse the energy conditions, stability, and amount of
exotic matter in this space-time. Next, as an alternative source in a simple
$f(R)$ gravity model, we use the background cloud of strings to estimate the
wormhole shape function, and analyse the relevant properties of the space-time.
These results are then compared with those of wormholes threaded by normal
matter in the simple $f(R)$ gravity model considered. The results demonstrate
that wormholes with NEC violations are feasible; however, the wormhole
space-times in the simple $f(R)$ gravity model are unstable.
