In what is a notable double deal, it appears the Yellow Submarine have acquired both players for a combined fee that is closer to €10m than it is to €20m.
Parejo will sign for three years at Villarreal with Coquelin penning a four-year deal, with the relatively low transfer fee allowing both players to keep their current salaries.
The deal is not yet official but is expected to be confirmed early this week, with Los Che set to return to training on Wednesday ahead of pre-season and will be without both players.
The two midfielders were said to be among the 12 first-team players at the club who were told they would be allowed to move clubs this summer amid a huge squad clearout at Los Che.
Coquelin has made 89 first-team appearances for Los Che since his January 2018 switch from the Gunners.
Parejo, 31, joined Valencia from Getafe in 2011 in a €6m deal and he has amassed 383 appearances for the club, netting 64 goals in that timeframe and becoming renowned as a set-piece expert.
It comes in a week after reports claimed Villarreal will sign Real Madrid teen Takefusa Kubo on a loan deal, with the club keen to bolster their midfield following the exits of Santi Cazorla and Bruno Soriano this summer.
New Villarreal boss Unai Emery is said to be intent on reinforcing his squad early in the window.