He will depart after spending 21 years at the north London club, with James Maddison set to head the other way in a separate deal.
Earlier in the week it was reported that new Spurs boss Ange Postecoglou is keen to offload up to ten players this summer.
One of the names suggested to depart was Winks, who spent last season on loan at the recently relegated Serie A side Sampdoria.
Although his Spurs career appears to be coming to an end, the 27-year-old now looks set to join his new club.
Winks is close to completing a £10million move to Leicester ahead of the new season.
The Foxes will be competing in the Championship next season following their relegation from the Premier League.
Having made 223 competitive appearances for Spurs and Sampdoria, he will bring a depth of experience to a side who will seek an instant return to the top flight.
Winks' move will not be the only deal between Leicester and Spurs as Maddison looks set to finalise a move to the latter.
The England international will have his medical on Wednesday after a £40million deal was agreed.
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Newcastle had also shown an interest in Maddison, but Spurs are said to have been his preferred destination this summer.
Both moves signal the start of the clubs' respective rebuilds as they make preparations for the 2023/24 campaign.
Leicester, now managed by Enzo Maresca, are likely to be one of the favourites to win the Championship title this coming season.
Back in the Premier League, Spurs will be eyeing qualification for Europe after missing out on continental football with their 8th placed finish.