He could not comment on whether the incident might jeopardise the player's role in the team.
The union would wait until the court process was completed before any employment procedure commenced, Mr Kinley said.
That would inevitably involve the New Zealand Rugby Players' Association, he told the Otago Daily Times.
It is not Morris' first time before the court.
According to ODT archives, the defendant was convicted of drink-driving as a 19-year-old in June 2013.
His breath-alcohol level on that occasion was 668mcg.
Morris was fined $650 and banned from driving for six months.
The news of the new drink-driving charge comes just days after the player revealed he would be moving to the United States in 2021.
Morris has signed a two-season contract to play for the New England Free Jacks in Major League Rugby.
There he will link up again with his former Otago Boys' High School coach Ryan Martin, who was thrilled by the signing.
"Aleki is a super-electric midfield back who possesses the ability to tear opposition defences apart with his fend-speed combo," Mr Martin said.
He called Morris "a brutal defender who loves big contacts and reads the game incredibly well".
The defendant is on bail until his next appearance in October.
Morris was contacted for comment.
