The grey squirrel, which originated in the US, is common in the UK but on the continent currently only has a foothold in Italy.
Until recently the slowly expanding small populations of Italian greys have posed little threat.
But new research shows that genetically distinct groups of the squirrels are now at the point of merging. Scientists fear this would greatly accelerate their ability to thrive in new environments.
If the greys manage to cross the Alps, it could prove disastrous for native red squirrels in other parts of Europe.
In the UK, red squirrel numbers collapsed as a result of grey squirrels out-competing them and spreading the deadly squirrel pox virus.
Grey squirrels have also changed the composition of British forests, by debarking trees such as Garry oaks.
Dr Lisa Signorile, from the Department of Life Sciences at Imperial College London, said: "Italian grey squirrels are edging closer to the northern border and are perilously close to crossing the Alps. If the Italian populations interbreed, they will increase in genetic diversity, which will increase their chances of invading the rest of Europe.
"To stop the spread, we need to understand what makes some populations such successful invaders.Our study helps us uncover some of those reasons."
The team compared 12 DNA markers from grey squirrels in Piedmont, northern Italy, with the same markers from greys in Northern Ireland, Northumberland and East Anglia.