Aina will join Scott Parker's side on loan with an £11m option to buy, while Tete will join on a permanent deal for £3m.
Both players will now finalise personal terms ahead of completing moves to Craven Cottage.
Netherlands international Tete has won 13 caps for his country and came through the Ajax youth academy having joined aged 10.
The 24-year-old made 55 appearances for Ajax, helping them reach the Europa League final in 2017, before moving to Lyon for £3.6m that summer.
He was an unused substitute in the Netherlands' 1-0 home defeat to Italy in the Nations League on Monday.
Former England youth international Aina is a graduate of Chelsea's academy and spent over a decade at the club.
He spent the 2017/18 campaign away on loan at Hull before making a season-long loan move to Torino the following year.
The Serie A side took up the option to make Aina's move from Chelsea permanent in June 2019. The 23-year-old made 37 appearances for Torino last season.
Fulham have made three close-season acquisitions following their return to the Premier League, with goalkeeper Alphonse Areola arriving from Paris Saint-Germain, Mario Lemina joining on a season-long loan and Harrison Reed signing on a permanent basis after spending last season at Craven Cottage.