The team has designed four potential options, and the winner of the public vote will be used on its cars at the Singapore, Japanese and Qatar Grands Prix.
Gulf was keen to repeat its popular one-off arrangement with McLaren at the 2021 Monaco GP, but with the vote and a different set of races adding a twist.
The four liveries are called Heritage, Contemporary, Visionary and Bolder than Bold.
The first is reminiscent of that used by McLaren and is intended to reflect Gulf's early years with Ford and Porsche in sportscar racing, while the other three are more modern interpretations.
Williams team boss James Vowles has indicated that he prefers the version that reflects Gulf's history.
"That's an era that I look back to in motorsport as being just an excellent time," said Vowles. "It's about racing more than anything else. And the Gulf livery, the GT40 and Porsche after that, is just iconic.
"And so to be partnered up with Gulf, and have the opportunity to do exactly the same thing with Williams, I'm excited by that.
"It just represents our sport. It represents our history. And we have two brands with immense legacy coming together. I think that's why the colours work. It's not the colours themselves, it's the nature of the legacy that they have behind them."
Vowles says the more modern takes on the Gulf colours are meant to reflect a change in F1's demographics.
"There is a whole range of audience really that is now in F1," he said. "So you have the iconic brand that's in there that represents the history.
"But you also have real modern twists on that in varying forms. And I think that's really a demographic that is now in F1.
"And that's what I appreciate about F1. F1 has grown tremendously across the last 10 years. If you look at the demographic, it's not now just an ageing population. The growth is actually within women, and a young age group.
"And [the liveries] are quite different, but quite similar. It's hard to explain, but they are modern I would say in their interpretation, in different ways."
Williams driver Alex Albon has indicated that, like his team boss, he prefers the traditional look.
"The Heritage for me is great," he said. "I'm a simple man, I like clean lines. I used to like the business, even race helmets I used when I was a young kid I used to love all the all the swivels and the gradient changes and all this kind of thing.
"And I've transitioned to basically a monotone helmet. I've matured!"
To vote, visit Williams' website.