Jacques Villeneuve, the 1997 world champion who was the last Canadian to race for the British-based team, told Reuters at Monza that he believed the deal was all but done for the 17-year-old.
A team source confirmed separately that Stroll was on the short list, along with Brazilian Felipe Nasr, who was a Williams reserve before securing his current race seat at Sauber.
Stroll turns 18 in October and is on course to win the European F3 title, which would secure him the necessary super-licence.
His career has been bankrolled by billionaire father Lawrence, who made his fortune with the Tommy Hilfiger and Michael Kors fashion brands.
Williams signed the youngster last year as a development driver with a promise he would be 'fully immersed' in the team in the same way that current race driver Valtteri Bottas was.
Mexican Sergio 'Checo' Perez has also been in the rumour mill but Force India, who are battling Williams for fourth place, have him under contract.
"We're just going through the process now, which is very close, to finalising the commercial side and then hopefully there will be an announcement in the near future," deputy principal Bob Fernley told Reuters.
"At the moment, his alternatives are either sideways or backwards."
Perez told reporters he hoped for an announcement within weeks and was confident he would get "what I always wanted".
When asked what that might be, he replied "You will find out very soon."
Some interpreted that as a hint that he could move to Renault, with that works team rebuilding, but it could equally refer to release clauses.
Ferrari have Kimi Raikkonen, the 2007 champion and oldest driver on the starting grid at 36, out of contract at the end of 2017 and Perez started out as a member of the Ferrari young driver academy.
The Mexican, who had one unhappy season at McLaren, has made no secret of his desire to drive for a top team again and being locked into a multi-year contract would make that harder.
The fate of Denmark's Kevin Magnussen and British rookie Jolyon Palmer at Renault remains uncertain, with the latter yet to score a point in 14 races.
Russian Daniil Kvyat is also looking insecure at Toro Rosso, with French GP2 leader Pierre Gasly eager for a chance.
(Reporting by Alan Baldwin, editing by Clare Lovell)
