Jim Duffy’s Championship outfit are due to play at Parkhead tomorrow, with a 12:30pm kick-off.
The holders have been pulling all out the stops to ensure the areas surrounding the ground is clear of snow and ice.
But Cappielow supremo Hawke insists the game should not go ahead.
He said: “We will have supporters risking life and limb to travel to a football match while the Met Office is issuing warnings to avoid non-essential travel.
“Ultimately, it isn’t about the area around Celtic Park.
“Supporter safety has to be key and I don’t think anyone can guarantee that at the moment and I fear for a lot of Morton supporters trying to get to this game .
“Surely, the Scottish Football Association and Police Scotland will come together and make a sensible early decision in the best interests of the safety of everyone.”
Aberdeen’s clash with Kilmarnock at Pittodrie became the first tie to be confirmed as being on.