A measurement of the underlying activity in events with a jet of transverse momentum in the several GeV region is performed in proton-proton collisions at root s = 0.9 and 7 TeV, using data collected by the CMS experiment at the LHC. The production of charged particles with pseudorapidity vertical bar eta vertical bar < 2 and transverse momentum p(T) > 0.5 GeV = c is studied in the azimuthal region transverse to that of the leading set of charged particles forming a track-jet. A significant growth of the average multiplicity and scalar-p(T) sum of the particles in the transverse region is observed with increasing p(T) of the leading track-jet, followed by a much slower rise above a few GeV/c. For track-jet p(T) larger than a few GeV/c, the activity in the transverse region is approximately doubled with a centre-of-mass energy increase from 0.9 to 7 TeV. Predictions of several QCD-inspired models as implemented in PYTHIA are compared to the data.