Spurs closed the gap to Premier League-leaders Chelsea to seven points as goals from Eric Dier and Heung-min Son sealed a 2-0 win.
But the victory at Turf Moor was soured by injuries to Victor Wanyama and Winks, with the latter stretchered from the field and taken to hospital.
Standard Sport understands Winks returned to London with the rest of the Tottenham team on Saturday evening after leaving hospital, and will be assessed on Monday.
Speaking after the game, Pochettino told Tottenham's official website: "It’s a problem in his ankle
"Now, we need to wait for results. We hope it won’t be a big issue but we’ll see. It’s sad, always. Harry is very young and an important player for us and it’s always bad news when a player gets an injury.