OK, we've got three problems here: First, in the title of your question, you are asking about "a gravity assist outside the solar system" but in the body of the question you ask about passing close to the Sun from within the Solar system. I think the title of this question should be edited to be "Could the Sun be used as a gravity assist to acheive an escape from the Solar system (with current tech)?"
The second problem is with PearsonArtPhoto's accepted answer wherein the Sun is assumed to be a stationary object but in fact it is in orbit around the center of the galaxy. Therefore, swinging around the Sun in the direction of its orbit will produce exactly the same sort of gravity assist as going past Jupiter (or any other relatively massive object).
The third problem is that a gravity assist from the Sun is not required to attain escape velocity from the Solar system - we already have FOUR probes that are leaving, never to return - two Pioneers (10 and 11, launched in 1972 and 1973), and two Voyagers (launched in 1977).
Pioneer 10's last, very weak signal was received on 23 January 2003. NASA engineers calculated its radioisotope power source has decayed to where it does not have enough power to send additional transmissions to Earth.
Pioneer 10 will continue to coast silently as a ghost ship through deep space into interstellar space, heading generally for the red star Aldebaran, which forms the eye of Taurus (The Bull). Aldebaran is about 68 light years away and it will take Pioneer over 2 million years to reach it.
Pioneer 11 studied energetic particles in the outer heliosphere.
The Pioneer 11 Mission ended on 30 September 1995, when the last transmission from the spacecraft was received. There have been no communications with Pioneer 11 since. The Earth's motion has carried it out of the view of the spacecraft antenna. The spacecraft cannot be maneuvered to point back at the Earth. It is not known whether the spacecraft is still transmitting a signal. No further tracks of Pioneer 11 are scheduled. The spacecraft is headed toward the constellation of Aquila (The Eagle), Northwest of the constellation of Sagittarius. Pioneer 11 will pass near one of the stars in the constellation in about 4 million years.
Voyager 1 is already in interstellar space, and Voyager 2 is in the heliosheath, and both spacecraft are still sending scientific information about their surroundings through the Deep Space Network.