The midfielder only joined the Toffees from Tottenham in January on a two-and-a-half-year deal as one of the first signings for manager Frank Lampard.
Alli made 11 appearances for Everton as they escaped from Premier League relegation last season, with only one start, and has featured twice so far in the new campaign, both times from the bench.
The 26-year-old was an unused substitute in the 1-1 draw against Nottingham Forest at Goodison Park last weekend, and will now look to get his career back on track with Besiktas.
Everton signed Alli on what was essentially a free transfer, but would have owed Spurs £10million once he had played 20 games across all competitions. That fee could have potentially risen to £40m if performance-related add-ons were met.
Everton is paying a portion of Alli's wages for the duration of his loan period in Istanbul.
While Alli will hope to play regular football again, the move abroad likely ends any distant chances he may have had of forcing his way back into Gareth Southgate's England squad for the Qatar 2022 World Cup.
The attacking midfielder, who has not scored an open play goal in the Premier League since January 2020, was a key member of the England team that reached the semi-finals at Russia 2018, scoring in the 2-0 quarter-final win against Sweden, but was omitted from last year's squad for Euro 2020.