Press Association Sport understands the matter has been handled amicably and will allow Law to return home to the United States for family reasons.
Law is a close ally of Gunners boss Arsene Wenger and has been working in the transfers department at Arsenal for the past eight years.
The club appointed Huss Fahmy, formerly of cycling outfit Team Sky, in June to work alongside Law - chiefly on player contracts.
It is understood that a decision has yet to be taken on whether Fahmy will now fulfil the role in Law's absence or if the club will look to recruit a direct replacement.
While Wenger has come in for criticism following a transfer window that left many Arsenal fans disappointed, even though they broke their transfer record to recruit Alexandre Lacazette, Law has also faced condemnation.
Arsenal's only other new addition alongside France striker Lacazette was the free signing of defender Sead Kolasinac.
The club hung on to Alexis Sanchez despite late bids from Manchester City as a deal to replace the Chilean with Monaco's Thomas Lemar broke down.
But they still sold Alex Oxlade-Chamberlain to Premier League rivals Liverpool and cleared out plenty of other fringe players.
It has also been reported that chief executive Ivan Gazidis has opted to take an office at Arsenal's London Colney training base.
Meanwhile the London Evening Standard claims Gazidis is set to meet with a number of influential fans ahead of Saturday's match at home to Bournemouth.
There are already plans being drawn up by some supporters to protest against what they consider to be an unsatisfactory transfer window, with apathy hastened by two defeats from their opening three Premier League games of the season.
Source : PA