The Voyagers are moving in the general direction of the centre of our Milky Way Galaxy
Scientists working on Voyager 1 are receiving further data suggesting the probe is close to crossing into interstellar space.
The Nasa mission, which launched from Earth in 1977, could leave our Solar System at any time.
It is now detecting a sharp rise in the number of high-energy particles hitting it from distant exploded stars.
The observation was predicted, and is another indication that Voyager will soon reach its historic goal.
"The laws of physics say that someday Voyager will become the first human-made object to enter interstellar space, but we still do not know exactly when that someday will be," Ed Stone, the Voyager project scientist at the California Institute of Technology in Pasadena, said in a Nasa statement.
"The latest data indicate that we are clearly in a new region where things are changing more quickly. It is very exciting. We are approaching the Solar System's frontier."
Voyager 1 is travelling at about 17 km per second (38,000 mph), and is almost 18 billion km (11 billion miles) from Earth.
The vast separation means a signal from the probe takes more than 16 and a half hours to arrive at Nasa's receiving network.
In the last three years, Voyager has seen a steady increase in the number of cosmic rays entering its two high-energy telescopes, but in the past month the counts have jumped markedly.
The cosmic ray count is one of three indicators Nasa is using to determine when the probe has moved to interstellar space.
The second is a change in the intensity of the energetic particles Voyager detects around it coming from our Sun.
The number of these hits is declining, but not dramatically so, which should happen when Voyager leaves the region of space dominated by our star.
A third indicator will be a change in the direction of the magnetic field lines. These are expected to undergo a major reorientation when Voyager breaks into interstellar space.
Voyager 1 was launched on 5 September 1977, and its sister spacecraft, Voyager 2, on 20 August 1977.
The probes' initial goal was to survey the outer planets Jupiter, Saturn, Uranus and Neptune, a task they completed in 1989.
They were then despatched towards deep space, in the general direction of the centre of our Milky Way Galaxy.
Their plutonium power sources will stop generating electricity in about 10-15 years, at which point their instruments and transmitters will die.
The Voyagers will then become "silent ambassadors" from Earth as they move through the Milky Way.
Voyager 1 is on course to approach a star called AC +793888, but it will only get to within two light-years of it.
Voyager 2 was launched before Voyager 1 and was put on a slower path to interstellar space. It is currently 14.7 billion km from Earth.
It is hurtling towards a star named Ross 248, but, again, even at its closest, it will still be a whole light-year away.
The domain of the Sun's influence is called the heliosphere: The Voyagers are approaching the edge of this enormous balloon of charged particles thrown out into space by our star
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