Robert Kubica and Sergey Sirotkin are the two names most heavily linked with the seat alongside Lance Stroll, with recent media speculation suggesting Sirotkin has edged ahead of Kubica. Daniil Kvyat, Paul di Resta and Pascal Wehrlein have also been connected with the drive.
Both Kubica and Sirotkin tested for the team in Abu Dhabi at the end of the season, although on different run plans - Kubica ran the hypersoft tyre for his quickest time, while Sirotkin's best, just half a second down, was set on the softs.
Williams have been clear that they will not rush their decision however - and with the traditional winter break on the horizon, they confirmed they will now wait for January before revealing their new line-up.
Kubica, 33, is bidding to complete what would be a remarkable comeback from life-threatening injuries suffered in a rally crash in February 2011. The 2008 Canadian Grand Prix winner competed in the WRC in the intervening years, but this year stepped up activities in F1, totalling more than 1,300 kilometres in tests with Renault and Williams.
Sirotkin meanwhile came close to breaking into F1 with Sauber back in 2014. The 22-year-old is a three-time racewinner in the newly-named F2 championship, and has served as Renault's F1 test and reserve driver for the past two years.