However, Dundee say the Scottish Football League have refused them permission for that arrangement.
"Unfortunately they turned us down for reasons of league integrity and us gaining an unfair advantage," said Dundee chief executive Harry McLean.
Griffiths joined Dundee from Livingston in the summer of 2009, and has scored 29 times for the Tayside outfit.
That form by the 20-year-old also earned him call-ups to the Scotland under-21 squad
"We approached the SFL hoping they would take a commonsense approach - because we're not allowed to sign anyone," McLean told BBC Scotland.
If there's a sensible bid then the administrator will decide. The chances are Leigh will have to leave the club in the very near future
Dundee chief executive Harry McLean
"The purpose for the sale is not to gain any advantage. It purely gives the administrators some money to help the club get out of administration.
"This solution would have done that and maintained our squad of 12 players to give a game on a Saturday to every team in the league.
"We'd have thought that keeping one of Scotland's finest young talents within the league would get good exposure.
"It was taken to the SFL board for discussion and it came back negative.
"If there's a sensible bid then the administrator will decide. The chances are Leigh will have to leave the club in the very near future."
Dundee are seeking fresh investment to help them get out of administration, where they have been since October.
At the beginning of January, their administrator Bryan Jackson warned that the First Division club could yet go out of business, with the tax authorities holding the key to their future existence.
The tax man is seeking £550,000 as the First Division club agreed to pursue a creditors voluntary arrangement in a bid to come out of administration.
Source: BBC Sport