Goal revealed earlier on Wednesday that Depay had sealed a switch to the Ligue 1 club, ending a nightmare spell at Old Trafford.
And Aulas has now stated his optimisim that a deal can be finalised with just the finer details to be hammered out.
The official also suggested the numbers involved could be in excess of those being widely speculated – namely a transfer fee that could eventually rise to €25 million.
"I got an agreement in principle early in the afternoon with Manchester United that remains to be detailed but on a basis higher than the circulating information," Aulas told L'Equipe.
Depay inspired PSV to the Eredivisie title in 2014-15 but did not maintain his stellar form under his former Netherlands boss Louis van Gaal at United last season, scoring twice in 29 Premier League appearances.
The 22-year-old has almost entirely fallen out of the first-team picture with Jose Mourinho at the helm this time around.
His solitary start came against Northampton Town in the EFL Cup and the last of four brief substitute cameos in the Premier League came in the 0-0 draw versus Burnley in October.
Lyon's official Twitter account started following Depay on Wednesday and he is poised to join the club fourth in Ligue 1 – 11 points behind leaders Monaco and Nice.
Bruno Genesio, Lyon's head coach, last week labelled Depay as his club's priority transfer target for the January window, while Mourinho said United were waiting for a "reasonable" bid that now appears to have arrived.