The rugby-league convert is gradually getting used to life on the big stage after making his England debut against France - a match that saw him feed Ben Foden for the game's only try but also butcher a clear try-scoring opportunity later in the match.
But Greenwood - who stands behind only Rory Underwood as England's all-time leading try-scorer with 31 tries - is convinced it is only a matter of time before Ashton starts making his mark at international level.
He said: "Chris Ashton has only been around union for a short time but he has had a massively impressive learning curve.
"He reminds me of Gary Lineker in football who just always knew where the net was - it was the same with Rory Underwood and Joe Roff who both possessed that killer instinct.
"There are only so many things you can coach, but he has a gift that you can't teach and that is the natural ability to score tries.
"I think the England management were waiting with Chris to see him cut out some little errors but that said I would have given him more game time in the Six Nations.
"He went to France and had that chance which he butchered. I was speaking to him about it and it is still burning inside him but that is a good sign. Choosing the chip instead of turning on the gas is something that will haunt him for the next 20 years.
"When I look back on my career I don't remember the 31 tries I scored for England, I think about the chances I blew in Cape Town and Dublin.
"It eats away inside you and it will give him all the motivation he needs the next time he is in an England shirt."
It is not just Ashton who has impressed this season at Franklins Gardens with fellow Northampton starlets Courtney Lawes and Foden also getting their first taste of Six Nations action while the likes of Phil Dowson, Shane Geraghty and Jon Clarke are all knocking at the door of the English management.
An Amlin Challenge Cup victory last season and EDF Energy Cup this campaign are proof that the Saints are on the march under Jim Mallinder.
But it was their performance in the 19-3 defeat of arch-rivals Leicester in February that convinced ex-Tiger Greenwood that the Saints will be eating at the top table of English rugby for some time to come.
He said: "They had a cracking win at Franklins Gardens and they out-Leicestered the Tigers.
"They bullied them off their own ball and tore them apart.
"I feel Leicester will still win the title this season but Northampton have a great owner, stadium and very talented squad controlled by some outstanding coaches.
"They will be at the top of English rugby for the next four to five years and it is only a matter of time before they win a Premiership or Heineken Cup."