Williams are in the market for a new driver this winter following Felipe Massa's decision to bring down the curtain on his Formula One career.
And Robert Kubica is the man hotly-tipped to join their roster alongside Stroll, who's on course to end his debut season 11th in the drivers' standings.
Kubica has been lined up for a test drive at Abu Dhabi's Yas Marina Circuit after the final race of the campaign on Sunday November 26.
Williams co-founder Head has now revealed Stroll had set his sights on being partnered by 2017 F1 champion Hamilton though.
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Lance was asked himself who he would like to see and he jokingly said Lewis Hamilton
Sir Patrick Head reveals who Lance Stroll wanted to replace Felipe Massa
"The big difficulty is, does he still have the toughness and heart for it? Only he knows that. And is he physically capable? He's certainly got a very heavily damaged arm and Williams obviously want to find that out.
"If he is physically capable, he will be a very good choice.
"Lance was asked himself who he would like to see and he jokingly said Lewis Hamilton.
Lewis Hamilton wrapped up his fourth Formula One title at the Mexican Grand Prix
Lance Stroll jokingly told Williams he wanted Lewis Hamilton to replace Felipe Massa
"Obviously this has been a big learning year for Lance and in order for him to be rated, then the chap in the other car needs to be highly rated as well. You are always rated against your team-mate.
"For Lance, his view will be he needs the strongest driver possible in the other car.
"There's always going to be question marks about Robert because of his physical limitations until he proves that's not the case. Maybe a test in Abu Dhabi will show internally if he has the capability."
Williams reserve driver Paul Di Resta has also been touted as a potential replacement for the retiring Massa.