If we assume that the asteroid is the equivalent of the Chicxulub impactor (so not quite big enough to kill all life on earth, but probably enough to wipe out humanity), aimed at the centre of the Earth (for working out the worse-case scenario), and that the Earth's gravity had no effect on the asteroid (for ease of working it out), we wouldn't be able to do anything about it.
This asteroid would be 15km wide, and if we assume that it is roughly spherical (for ease of calculation...see jokes about spherical cows) and has a density of 2000 kg/m3, the mass would be 1.77 trillion tonnes. If we could act instantly, we would have 5 days to deflect it by about 7000 km, so 1400 km/day or 16.2 m/s. The energy required to move mass m (in kg) at velocity _v (in m/s) is at least 1/2mv2. For 1.77 trillion tonnes (=1.77 x 1015 kg) by 7000km in 5 days, it would require 4.645 x 1017 Joules, equivalent to about 110 megatonne explosion, assuming we can focus all of that energy onto the asteroid rather than simply having it throw out equally in all directions.
However most of the energy from such an explosion would probably break up the asteroid into smaller fragments, rather than moving it, so resulting in a rain of fiery asteroid fragments impacting the Earth, rather than a single large one. It could be done with rockets pushing it, in theory, but that's out of the question (literally), so I'm not sure we can put up the necessary amount of energy in the right spot.