Fronted by Chelsea fan Mohamed Al Khereiji, it was one of six groups who said it had lodged a bid to buy the Premier League club last week.
Sources close to the SMG bid said its desire to do due diligence on the club and Chelsea’s eagerness for a quick sale counted against it.
SMG offered more than £2bn, plus over £200m for immediate working capital.
It was informed it had not made the next stage on Thursday.
Some of the other bids have already done due diligence based on their previous interest in buying the club.
The Premier League club has been granted a special licence by the UK government to continue operating after owner Roman Abramovich was sanctioned for his alleged links to Russian president Vladimir Putin.
They are unable to receive funds from ticket sales and merchandise, and have a monthly wage bill of about £28m.
On Wednesday, the government granted permission for Abramovich’s company which owns Chelsea, Fordstam Ltd, to pay £30m into the club for running costs.
The Russian billionaire is seeking £3bn for the sale, but proceeds will go to a charitable foundation or a frozen account with approval from the government.
It is understood that bids are being assessed on their valuation of the club, their working capital commitment, their source of funds, the speed and certainty with which they can close the transaction and their background in sport.
Saudi Media Group says that its money is privately funded by one of the country’s biggest media outlets, and is not linked to the Saudi state. Al Khereiji is the chief executive of Saudi Media’s parent company, Engineer Holding Group.
The five others to have gone public with bids include a consortium involving the Ricketts family, who own the Chicago Cubs, a group involving Los Angeles Dodgers co-owner Todd Boehly and Swiss billionaire Hansjorg Wyss, plus another which includes former Liverpool chairman Sir Martin Broughton and Lord Coe.
Other consortiums include British property investor Nick Candy, who increased his initial £2bn bid, and global investment firm Centricus. Reports have suggested there has also been interest from New York Jets owner Woody Johnson.
Those bids are still waiting to hear whether they have made the shortlist from the American investment firm handling the sale, Raine Group.
BBC