Images and low-resolution spectra of the near-Earth Jupiter family comet (JFC) 249P/LINEAR in the visible range obtained with the instrument OSIRIS in the 10.4 m Gran Telescopio Canarias (GTC) (La Palma, Spain) on January 3, 4, 6 and February 6, 2016 are presented, together with a series of images obtained with the 0.4m telescope of the Great Shefford Observatory obtained on Oct. 22 and 27, and Nov. 1 and 24, 2006. The reflectance spectrum of 249P is similar to that of a B-type asteroid. The comet has an absolute (visual) nuclear magnitude HV = 17.0 ± 0.4 , which corresponds to a radius of about 1-1.3 km for a geometric albedo ∼ 0.04 - 0.07 . From the analysis of GTC images using a Monte Carlo dust tail code we find that the time of maximum dust ejection rate was around 1.6 days before perihelion. The analysis of the dust tails during the 2006 and 2016 perihelion approaches reveals that, during both epochs, the comet repeated the same dust ejection pattern, with a similar short-lived activity period of about 20 days (FWHM) around perihelion and a dust loss rate peaking at 145 ± 50 kg/s. The total dust mass ejected during its last perihelion passage was (2.5 ± 0.9) × 108 kg, almost all this mass being emitted before the first observation of January 3, 2016. The activity onset, duration, and total ejected mass were very similar during the 2006 perihelion passage. This amount of dust mass is very low as compared with that from other active JFCs. The past orbital evolution of 249P and 100 clones were also followed over a time scale of ∼ 5 × 104 yr. The object and more than 60% of the clones remained bound to the near-Earth region for the whole computed period, keeping its perihelion distance within the range q ≃ 0.4 - 1.1 au. The combination of photometric and spectroscopic observations and dynamical studies show that the near-Earth comet 249P/LINEAR has several peculiar features that clearly differentiate it from typical JFCs. We may be in front of a new class of near-Earth JFC whose source region is not the distant trans-neptunian population, but much closer in the asteroid belt. Therefore, 249P/LINEAR may be a near-Earth counterpart of the so-called main-belt comets or active asteroids.