The exomoon, which is a satellite that orbits a planet outside our solar system, appears to be revolving around the Jupiter-sized gas giant Kepler-1625b.
The exomoon itself is believed to be around the size of Neptune, which is about four times bigger than earth.
There are no moons anywhere near as big in our solar system.
The scientists admitted their findings are not definitive, however, as with any new discovery in space.
One of the astronomers, David Kipping, said: “We’ve tried our best to rule out other possibilities such as spacecraft anomalies, other planets in the system or stellar activity, but we’re unable to find any other single hypothesis which can explain all of the data we have.”
He added: “We hope to re-observe the star again in the future to verify or reject the exomoon hypothesis.
“And if validated, the planet-moon system – a Jupiter with a Neptune-sized moon – would be a remarkable system with unanticipated properties, in many ways echoing the expected discovery of hot Jupiters in the early days of planet hunting.”
The breakthrough was made when Assistant Professor Kipping and Graduate Fellow Alex Teachey noticed intriguing anomalies in the observations they made of 284 Kepler-discovered-planets.
Mr Kipping said: “We saw deviations and wobbles in the light curve that caught our attention.”
The scientists made the initial observations on the Kepler Space Telescope and used their evidence to secure 40 hours time on the Hubble.
They noticed that after Kepler-162b crossed in front of its star there was another decrease in measurable brightness 3.5 hours late.
Mr Kipping said the moon was “trailing the planet like a dog following its owner on a leash”.
While both the planet and its moon are within the habitable zone of their star (Kepler 1625) it is thought that both are gas planets are therefore incapable of hosting life.
The planet and its moon seem to have a similar size ratio that of the Earth and the Moon.