Al Faraj, 40, a property developer in the Middle East has already passed the Premier League's fit and proper test.
Shadow culture secretary Ivan Lewis said the government should introduce a wider public interest test and allow regulators to apply a fit and proper persons test.
Al Fahim has been named Pompey chairman after passing the Premier League's fit and proper person test.
If the punishment stands, he may be in violation of the Football League's 'fit and proper person test'.
The report said that a "fit and proper persons test" should be introduced for all club owners and directors.
Glover's takeover was initially agreed last week but was subject to the RFL's final written confirmation that he had passed a fit and proper person test.
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Chainrai and any directors he wants to appoint must now pass the Premier League's fit and proper persons test, but there is every likelihood his tenure will be brief.
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Pompey announced on 21 July they have agreed in principle to a takeover by United Arab Emirates tycoon Sulaiman Al Fahim after the 32-year-old passed the Premier League's fit and proper person test.
"You could argue that one of the reasons that Mr Shinawatra has decided that it might be right to to sell is that it might be at some point he may fail our fit and proper persons test, " added Scudamore.
The group also called for a tightening up of the FA's Fit and Proper Persons Test for potential club owners, saying that debt plans should be taken into account, and that proposed takeovers should be rejected "where it is considered against the long-term interests of the club".
The United Arab Emirates businessman passed the Premier League 'fit and proper person' test in July but the ownership deal has yet to go through.
These suspicions will continue to dog the takeover deal, and there are still major issues of plurality as well as the "fit and proper person" test.
Al Fahim, 32, agreed to buy Pompey from Alexandre Gaydamak in May and became chairman in July, after passing the Premier League's fit-and-proper-person test.
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