  Observations by HAWC and Milagro have detected spatially extended TeV sources
surrounding middle-aged ($t\sim100-400 \,{\rm kyr}$) pulsars like Geminga and
PSR B0656+14, which have been named "TeV Halos", representing very extended TeV
pulsar wind nebulae (PWNe) powered by relatively old pulsars. A few more
HAWC-detected sources have been suggested to be TeV halo candidates. In this
paper, we search for possible GeV counterparts of three TeV halo candidates
with Fermi Large Area Telescopes. We detect a new spatially extended GeV source
in the vicinity of the TeV halo candidate 2HWC J1912+099, which is also
detected by HESS (HESS J1912+101). We find that the size of the GeV source is
significantly larger than that of the TeV emission measured by HESS, and a
spatial template characteristic of a PWN can fit the GeV data. We suggest that
the GeV source is an extended PWN powered by the central middle-aged pulsar PSR
J1913+1011. This discovery favors the TeV halo scenario for the TeV source 2HWC
J1912+099 (HESS J1912+101), although the possible shell-like morphology
measured by HESS challenges this interpretation. Alternatively, the TeV
emission could be dominated by a supernova remnant (SNR) via the hadronic
process. Future multi-wavelength observations of the source and more precise
measurements of the spatial profile of the TeV emission will be useful to
distinguish between the two scenarios.
